query: numbers that are the sum of two squares

pos: Fermat's theorem on sums of two squares even exponent in the prime factorization of the number. Fermat's theorem on sums of two squares In additive number theory, Fermat's theorem on sums of two squares states that an odd prime "p" can be expressed as: formula_1 with "x" and "y" integers, if and only if The prime numbers for which this is true are called Pythagorean primes. For example, the primes 5, 13, 17, 29, 37 and 41 are all congruent to 1 modulo 4, and they can be expressed as sums of two squares in the following ways: On the other hand, the primes 3, 7, 11,

pos: Sum of two squares theorem Sum of two squares theorem In number theory, the sum of two squares theorem says when an integer can be written as a sum of two squares, that is, when for some integers , . This theorem supplements Fermat's two-square theorem which says when a prime number can be written as a sum of two squares, in that it also covers the case for composite numbers. The prime decomposition of the number 2450 is given by 2450 = 257. Of the primes occurring in this decomposition, 2, 5, and 7, only 7 is congruent to 3 modulo 4. Its exponent

pos: Square number first odd integers, beginning with one, is a perfect square. 1, 1 + 3, 1 + 3 + 5, 1 + 3 + 5 + 7, etc. The sum of the "n" first cubes is the square of the sum of the "n" first positive integers; this is Nicomachus's theorem. All fourth powers, sixth powers, eighth powers and so on are perfect squares. Squares of even numbers are even (and in fact divisible by 4), since . Squares of odd numbers are odd, since . It follows that square roots of even square numbers are even, and square roots of

pos: Pythagorean prime Pythagorean prime A Pythagorean prime is a prime number of the form 4"n" + 1. Pythagorean primes are exactly the odd prime numbers that are the sum of two squares; this characterization is Fermat's theorem on sums of two squares. Equivalently, by the Pythagorean theorem, they are the odd prime numbers "p" for which is the length of the hypotenuse of a right triangle with integer legs, and they are also the prime numbers "p" for which "p" itself is the hypotenuse of a Pythagorean triangle. For instance, the number 5 is a Pythagorean prime; is the hypotenuse of a

pos: Antimagic square Antimagic square An antimagic square of order "n" is an arrangement of the numbers 1 to "n" in a square, such that the sums of the "n" rows, the "n" columns and the two diagonals form a sequence of 2"n" + 2 consecutive integers. The smallest antimagic squares have order 4. In each of these two antimagic squares of order 4, the rows, columns and diagonals sum to ten different numbers in the range 29–38. In the antimagic square of order 5 on the left, the rows, columns and diagonals sum up to numbers between 60 and 71. In the

pos: 50 (number) 50 (number) 50 (fifty) is the natural number following 49 and preceding 51. Fifty is the smallest number that is the sum of two non-zero square numbers in two distinct ways: 50 = 1 + 7 = 5 + 5. It is also the sum of three squares, 50 = 3 + 4 + 5, and the sum of four squares, 50 = 6 + 3 + 2 + 1. It is a Harshad number. There is no solution to the equation φ("x") = 50, making 50 a nontotient. Nor is there a solution to the equation "x" − φ("x")

pos: 50 (number) = 50, making 50 a noncototient. Fifty is: 50 (number) 50 (fifty) is the natural number following 49 and preceding 51. Fifty is the smallest number that is the sum of two non-zero square numbers in two distinct ways: 50 = 1 + 7 = 5 + 5. It is also the sum of three squares, 50 = 3 + 4 + 5, and the sum of four squares, 50 = 6 + 3 + 2 + 1. It is a Harshad number. There is no solution to the equation φ("x") = 50, making 50 a nontotient. Nor is there

pos: Nonhypotenuse number Nonhypotenuse number In mathematics, a nonhypotenuse number is a natural number whose square "cannot" be written as the sum of two nonzero squares. The name stems from the fact that an edge of length equal to a nonhypotenuse number "cannot" form the hypotenuse of a right angle triangle with integer sides. The numbers 1, 2, 3 and 4 are all nonhypotenuse numbers. The number 5, however, is "not" a nonhypotenuse number as 5 equals 3 + 4. The first fifty nonhypotenuse numbers are: Although nonhypotenuse numbers are common among small integers, they become more-and-more sparse for larger numbers. Yet, there

pos: Pell number any higher power of an integer are 0, 1, and 169 = 13. However, despite having so few squares or other powers, Pell numbers have a close connection to square triangular numbers. Specifically, these numbers arise from the following identity of Pell numbers: The left side of this identity describes a square number, while the right side describes a triangular number, so the result is a square triangular number. Santana and Diaz-Barrero (2006) proved another identity relating Pell numbers to squares and showing that the sum of the Pell numbers up to "P" is always a square: For instance, the

pos: Pandiagonal magic square main diagonals and broken diagonals have a sum of 150 and, if we square all the numbers in the square, only the rows and the columns are magic and have a sum of 5150. For 10th order a similar construction is possible using the equal partitionings of the sum 1+2+3+4+5+9+10+11+12+13 = 70: This leads to squares having a maximum element of 169 and a pandiagonal magic sum of 850, which are also semibimagic with each row or column sum of squares equal to 102,850. A formula_26 pandiagonal magic square can be built by the following algorithm. A formula_27 pandiagonal magic

pos: Square number application of the factorization of a difference of two squares.) For example, is the square of 3, so consequently divides 9991. This test is deterministic for odd divisors in the range from to where covers some range of natural numbers . A square number cannot be a perfect number. The sum of the "n" first square numbers is The first values of these sums, the square pyramidal numbers, are: 0, 1, 5, 14, 30, 55, 91, 140, 204, 285, 385, 506, 650, 819, 1015, 1240, 1496, 1785, 2109, 2470, 2870, 3311, 3795, 4324, 4900, 5525, 6201... The sum of the

pos: 25 (number) 25 (number) 25 (twenty-five) is the natural number following 24 and preceding 26. It is a square number, being 5 = 5 × 5. It is one of two two-digit numbers whose square and higher powers of the number also ends in the same last two digits, e.g. 25 = 625, the other is 76. It is the smallest square that is also a sum of two (non-zero) squares: 25 = 3 + 4. Hence it often appears in illustrations of the Pythagorean theorem. 25 is the sum of the single-digit odd natural numbers 1, 3, 5, 7 and 9,

pos: Antimagic square sparse totally anti-magic square (STAM). Note that a STAM is not necessarily a SAM, and vice versa. Antimagic square An antimagic square of order "n" is an arrangement of the numbers 1 to "n" in a square, such that the sums of the "n" rows, the "n" columns and the two diagonals form a sequence of 2"n" + 2 consecutive integers. The smallest antimagic squares have order 4. In each of these two antimagic squares of order 4, the rows, columns and diagonals sum to ten different numbers in the range 29–38. In the antimagic square of order 5 on

pos: Pandiagonal magic square of two addends: An additional equal partitioning of the sum of squares guarantees the semibimagic property noted below: Note that the consecutive integer sum 1+2+3+4+5+6 = 21, an odd sum, lacks the half-partitioning. With both equal partitions available, the numbers 1, 2, 3, 5, 6, 7 can be arranged into 6x6 pandigonal patterns and , respectively given by: Then formula_25 (where is the magic square with 1 for all cells) gives the nonconsecutive pandiagonal 6x6 square: with a maximum element of 49 and a pandiagonal magic sum of 150. This square is pandiagonal and semibimagic, that means that rows, columns,

pos: Proofs of Fermat's theorem on sums of two squares formula_29 and formula_30 are relatively prime positive integers then every factor of formula_3 is a sum of two squares." (This is the step that uses step (3.) to produce an 'infinite descent' and was Euler's Proposition 4. The proof sketched below also includes the proof of his Proposition 3). 5. "Every prime of the form formula_79 is a sum of two squares." (This is the main result of Euler's second paper). Lagrange completed a proof in 1775 based on his general theory of integral quadratic forms. The following presentation incorporates a slight simplification of his argument, due to Gauss, which

pos: Location arithmetic the value. This property can be used to perform binary addition using just a single row of the grid. First, lay out a binary number on a row using counters to represent the 1s in the number. For example, 29 (= 11101 in binary) would be placed on the board like this: The number 29 is clearly the sum of the values of the squares on which there are counters. Now overlay the second number on this row. Say we place 9 (= 1001 in binary) on it like this. The sum of these two numbers is just the total

neg: Fermat's theorem on sums of two squares modulo 4, and none of them can be expressed as the sum of two squares. This is the easier part of the theorem, and follows immediately from the observation that all squares are congruent to 0 or 1 modulo 4. Albert Girard was the first to make the observation, describing all positive integral numbers (not necessarily primes) expressible as the sum of two squares of positive integers; this was published in 1625. The statement that every prime "p" of the form "4n+1" is the sum of two squares is sometimes called "Girard's theorem". For his part, Fermat wrote an elaborate

neg: Landau–Ramanujan constant the same constant of proportionality, and with a slowly growing error term. Landau–Ramanujan constant In mathematics and the field of number theory, the Landau–Ramanujan constant is a number that occurs in a theorem stating that for large "x", the number of positive integers below "x" that are the sum of two square numbers varies as The constant is named after its discoverer, Srinivasa Ramanujan. By the sum of two squares theorem, the numbers that can be expressed as a sum of two squares of integers are the ones for which each prime number congruent to 3 mod 4 appears with

neg: Landau–Ramanujan constant Landau–Ramanujan constant In mathematics and the field of number theory, the Landau–Ramanujan constant is a number that occurs in a theorem stating that for large "x", the number of positive integers below "x" that are the sum of two square numbers varies as The constant is named after its discoverer, Srinivasa Ramanujan. By the sum of two squares theorem, the numbers that can be expressed as a sum of two squares of integers are the ones for which each prime number congruent to 3 mod 4 appears with an even exponent in the prime factorization. For instance, 45 = 9

neg: Pythagorean prime odd primes are 26861 and 26862. The sum of one odd square and one even square is congruent to 1 mod 4, but there exist composite numbers such as 21 that are 1 mod 4 and yet cannot be represented as sums of two squares. Fermat's theorem on sums of two squares states that the prime numbers that can be represented as sums of two squares are exactly 2 and the odd primes congruent to 1 mod 4. The representation of each such number is unique, up to the ordering of the two squares. By using the Pythagorean theorem, this

neg: Pandiagonal magic square 3 pandiagonal magic square does exist. The smallest non-trivial pandiagonal magic squares are squares. All pandiagonal magic squares must be translationally symmetric to the form Since each subsquare sums to the magic constant, pandiagonal magic squares are most-perfect magic square. In addition, the two numbers at the opposite corners of any square add up to half the magic sum. Consequently, all pandiagonal magic squares that are associative must have duplicate cells. All pandiagonal magic squares using numbers 1-16 without duplicates are obtained by letting equal 1; letting , , , and equal 1, 2, 4, and 8 in some order;

neg: Legendre's three-square theorem Legendre's three-square theorem In mathematics, Legendre's three-square theorem states that a natural number can be represented as the sum of three squares of integers if and only if is not of the form formula_2 for integers and . The first numbers that cannot be expressed as the sum of three squares (i.e. numbers that can be expressed as formula_2) are Pierre de Fermat gave a criterion for numbers of the form 3"a" + 1 to be a sum of three squares essentially equivalent to Legendre's theorem, but did not provide a proof. N. Beguelin noticed in 1774 that every positive

neg: Degen's eight-square identity Degen's eight-square identity In mathematics, Degen's eight-square identity establishes that the product of two numbers, each of which is a sum of eight squares, is itself the sum of eight squares. Namely: First discovered by Carl Ferdinand Degen around 1818, the identity was independently rediscovered by John Thomas Graves (1843) and Arthur Cayley (1845). The latter two derived it while working on an extension of quaternions called octonions. In algebraic terms the identity means that the norm of product of two octonions equals the product of their norms: formula_10. Similar statements are true for quaternions (Euler's four-square identity), complex numbers

neg: Carmichael number are infinitely many such R. The first Carmichael numbers with formula_29 prime factors are : The first Carmichael numbers with 4 prime factors are : The second Carmichael number (1105) can be expressed as the sum of two squares in more ways than any smaller number. The third Carmichael number (1729) is the Hardy-Ramanujan Number: the smallest number that can be expressed as the sum of two cubes (of positive numbers) in two different ways. Let formula_30 denote the number of Carmichael numbers less than or equal to formula_31. The distribution of Carmichael numbers by powers of 10 : In

neg: 4 has four vertices. Lagrange's four-square theorem states that every positive integer can be written as the sum of at most four square numbers. Three are not always sufficient; for instance cannot be written as the sum of three squares. Each natural number divisible by 4 is a difference of squares of two natural numbers, i.e. . Four is the highest degree general polynomial equation for which there is a solution in radicals. The four fours game, there are known solutions for all integers from 0 to 880 (but not 881). Representing 1, 2 and 3 in as many lines as

neg: Difference of two squares The denominator of formula_31 can be rationalised as follows: Here, the irrational denominator formula_38 has been rationalised to formula_39. The difference of two squares can also be used as an arithmetical short cut. If you are multiplying two numbers whose average is a number which is easily squared the difference of two squares can be used to give you the product of the original two numbers. For example: Which means using the difference of two squares formula_41 can be restated as The difference of two consecutive perfect squares is the sum of the two bases "n" and "n"+1. This can

neg: Square number have an even number of positive divisors. An integer root is the only divisor that pairs up with itself to yield the square number, while other divisors come in pairs. Lagrange's four-square theorem states that any positive integer can be written as the sum of four or fewer perfect squares. Three squares are not sufficient for numbers of the form . A positive integer can be represented as a sum of two squares precisely if its prime factorization contains no odd powers of primes of the form . This is generalized by Waring's problem. In base 10, a square number

neg: Hilbert's tenth problem versa. This can be seen as follows: The requirement that solutions be natural numbers can be expressed with the help of Lagrange's four-square theorem: every natural number is the sum of the squares of four integers, so we simply replace every parameter with the sum of squares of four extra parameters. Similarly, since every integer can be written as the difference of two natural numbers, we can replace every parameter that ranges in integers with the difference of two parameters that range in natural numbers. A recursively enumerable set can be characterized as one for which there exists an algorithm

neg: Octahedral number numbers "pyramides quadratae secundae". The number of cubes in an octahedron formed by stacking centered squares is a centered octahedral number, the sum of two consecutive octahedral numbers. These numbers are given by the formula Octahedral number In number theory, an octahedral number is a figurate number that represents the number of spheres in an octahedron formed from close-packed spheres. The "n"th octahedral number formula_1 can be obtained by the formula: The first few octahedral numbers are: The octahedral numbers have a generating function Sir Frederick Pollock conjectured in 1850 that every number is the sum of at most 7

neg: Most-perfect magic square were then scanned for squares with 20,15 in the proper cells for any of the 8 rotations. The 2015 squares all originated with principal reversible square number #31. This square has values that sum to 35 on opposite sides of the vertical midline in the first two rows. The image below shows areas completely surrounded by larger numbers with a blue background. A water retention topographical model is one example of the physical properties of magic squares. There are 108 of these 2x2 subsquares that have the same sum for the 4x4x4 most-perfect cube. Number shapes on a triangular grid

neg: Optic equation and "b" and half the length of the segment through the diagonal intersection as "c", the sum of the reciprocals of "a" and "b" equals the reciprocal of "c". The special case in which the integers whose reciprocals are taken must be square numbers appears in two ways in the context of right triangles. First, the sum of the reciprocals of the squares of the altitudes from the legs (equivalently, of the squares of the legs themselves) equals the reciprocal of the square of the altitude from the hypotenuse. This holds whether or not the numbers are integers; there is

neg: Euler's factorization method given formula_6 we find formula_7 as a product of sums of two squares. First deduce that and factor both sides to get Now let formula_10 and formula_11 so that there exists some constants formula_12 satisfying formula_15 formula_18 Substituting these into equation (1) gives Canceling common factors yields Now using the fact that formula_21 and formula_22 are pairs of relatively prime numbers, we find that So We now see that formula_29 and formula_30 Applying the Brahmagupta–Fibonacci identity we get As each factor is a sum of two squares, one of these must contain both even numbers: either formula_32 or formula_33. Without

neg: Brahmagupta of the squares and cubes of the first integers. 12.20. The sum of the squares is that [sum] multiplied by twice the [number of] step[s] increased by one [and] divided by three. The sum of the cubes is the square of that [sum] Piles of these with identical balls [can also be computed]. Here Brahmagupta found the result in terms of the "sum" of the first integers, rather than in terms of as is the modern practice. He gives the sum of the squares of the first natural numbers as and the sum of the cubes of the first n

neg: Hosoya's triangle the two outermost diagonals are the Fibonacci numbers, while the numbers on the middle vertical line are the squares of the Fibonacci numbers. All the other numbers in the triangle are the product of two distinct Fibonacci numbers greater than 1. The row sums are the first convolved Fibonacci numbers. Hosoya's triangle The Hosoya triangle or Hosoya's triangle (originally Fibonacci triangle) is a triangular arrangement of numbers (like Pascal's triangle) based on the Fibonacci numbers. Each number is the sum of the two numbers above in either the left diagonal or the right diagonal. The first few rows are: (See

neg: Scrabble variants or triplets of numbers that sum up to less than, or exactly 15, respectively. The placement of the pairs or triplets is similar to Scrabble. Triolet means "triplet" in French. Yushino uses number sequences where every number is the last digit of the sum of the previous two. For two number sequences it requires that the numbers are adjacent. Mixmath, a game developed by Wrebbit, also involves placing correct mathematical equations on a board with bonus squares similar to a scrabble board. Scrabble Upwords (originally just named UpWords) is played with 100 letter tiles on a special 10×10 board with

neg: Euler's four-square identity Euler's four-square identity In mathematics, Euler's four-square identity says that the product of two numbers, each of which is a sum of four squares, is itself a sum of four squares. For any pair of quadruples from a commutative ring, the following expressions are equal: Euler wrote about this identity in a letter dated May 4, 1748 to Goldbach (but he used a different sign convention from the above). It can be proven with elementary algebra. The identity was used by Lagrange to prove his four square theorem. More specifically, it implies that it is sufficient to prove the theorem

neg: Degrees of freedom (statistics) degrees of freedom. In some complicated settings, such as unbalanced split-plot designs, the sums-of-squares no longer have scaled chi-squared distributions. Comparison of sum-of-squares with degrees-of-freedom is no longer meaningful, and software may report certain fractional 'degrees of freedom' in these cases. Such numbers have no genuine degrees-of-freedom interpretation, but are simply providing an "approximate" chi-squared distribution for the corresponding sum-of-squares. The details of such approximations are beyond the scope of this page. Several commonly encountered statistical distributions (Student's t, Chi-Squared, F) have parameters that are commonly referred to as "degrees of freedom". This terminology simply reflects that in many applications

neg: Sum of two squares theorem in the decomposition, 2, is even. Therefore, the theorem states, it is expressible as the sum of two squares. Indeed, . The prime decomposition of the number 3430 is 257. This time, the exponent of 7 in the decomposition is 3, an odd number. So 3430 cannot be written as the sum of two squares. Sum of two squares theorem In number theory, the sum of two squares theorem says when an integer can be written as a sum of two squares, that is, when for some integers , . This theorem supplements Fermat's two-square theorem which says when a

neg: Degen's eight-square identity formula_17. Thus, where, and, with, Incidentally, the formula_32 obey the identity, Degen's eight-square identity In mathematics, Degen's eight-square identity establishes that the product of two numbers, each of which is a sum of eight squares, is itself the sum of eight squares. Namely: First discovered by Carl Ferdinand Degen around 1818, the identity was independently rediscovered by John Thomas Graves (1843) and Arthur Cayley (1845). The latter two derived it while working on an extension of quaternions called octonions. In algebraic terms the identity means that the norm of product of two octonions equals the product of their norms: formula_10.

neg: Magic square used to drive away melancholy. It is very similar to Yang Hui's square, which was created in China about 250 years before Dürer's time. The sum 34 can be found in the rows, columns, diagonals, each of the quadrants, the center four squares, and the corner squares (of the 4×4 as well as the four contained 3×3 grids). This sum can also be found in the four outer numbers clockwise from the corners (3+8+14+9) and likewise the four counter-clockwise (the locations of four queens in the two solutions of the 4 queens puzzle), the two sets of four symmetrical numbers

neg: Proofs of Fermat's theorem on sums of two squares of which is a sum of two squares, is itself a sum of two squares." 2. "If a number which is a sum of two squares is divisible by a prime which is a sum of two squares, then the quotient is a sum of two squares." (This is Euler's first Proposition). 3. "If a number which can be written as a sum of two squares is divisible by a number which is not a sum of two squares, then the quotient has a factor which is not a sum of two squares." (This is Euler's second Proposition). 4. "If

neg: Planar separator theorem the radii of the disks. However, the sum of the squares of the radii is proportional to the total area of the disks, which is at most the total surface area of the unit sphere, a constant. An argument involving Jensen's inequality shows that, when the sum of squares of "n" non-negative real numbers is bounded by a constant, the sum of the numbers themselves is O(√"n"). Therefore, the expected number of disks crossed by a random plane is O(√"n") and there exists a plane that crosses at most that many disks. This plane intersects the sphere in a great

neg: Proofs of Fermat's theorem on sums of two squares briefly sketched in the letter. The full proof consists in five steps and is published in two papers. The first four steps are Propositions 1 to 4 of the first paper and do not correspond exactly to the four steps below. The fifth step below is from the second paper. For the avoidance of ambiguity, zero will always be a valid possible constituent of "sums of two squares", so for example every square of an integer is trivially expressible as the sum of two squares by setting one of them to be zero. 1. "The product of two numbers, each

neg: Mandelbulb identity |v|=|v| to hold. More general fractals can be found by setting for functions f and g. Other formulae come from identities which parametrise the sum of squares to give a power of the sum of squares such as: which we can think of as a way to square a triplet of numbers so that the modulus is squared. So this gives, for example: or various other permutations. This 'quadratic' formula can be applied several times to get many power-2 formulae. Other formulae come from identities which parametrise the sum of squares to give a power of the sum of

neg: Idoneal number Idoneal number In mathematics, Euler's idoneal numbers (also called suitable numbers or convenient numbers) are the positive integers "D" such that any integer expressible in only one way as "x" ± "Dy" (where "x" is relatively prime to "Dy") is a prime, prime power, twice one of these, or a power of 2. In particular, a number that has two distinct representations as a sum of two squares is composite. Every idoneal number generates a set containing infinitely many primes and missing infinitely many other primes. A positive integer "n" is idoneal if and only if it cannot be written

neg: Parity problem (sieve theory) contain prime factors from formula_54, and as formula_61 a sebset of numbers from formula_59 with even number of prime factors, as formula_63 a sebset of numbers from formula_59 with odd number of prime factors. We define the functions Karatsuba proved that, for formula_66 the asymptotic formula formula_67 is valid, where formula_68 is a positive constant. He also showed that it is possible to prove the analogous theorems for other sets of natural numbers, for example, for numbers which are representable in the form of the sum of two squares, and that sets of natural numbers, all factors of which do

neg: Lack-of-fit sum of squares variables. For example, consider fitting a line by the method of least squares. One takes as estimates of "α" and "β" the values that minimize the sum of squares of residuals, i.e., the sum of squares of the differences between the observed "y"-value and the fitted "y"-value. To have a lack-of-fit sum of squares that differs from the residual sum of squares, one must observe more than one "y"-value for each of one or more of the "x"-values. One then partitions the "sum of squares due to error", i.e., the sum of squares of residuals, into two components: The sum

neg: Sun Zhiwei be represented as a sum of an even square and two triangular numbers. He conjectured, and proved with B.-K. Oh, that each positive integer can be represented as a sum of a square, an odd square and a triangular number. In 2009, he conjectured that any natural number can be written as the sum of two squares and a pentagonal number, as the sum of a triangular number, an even square and a pentagonal number, and as the sum of a square, a pentagonal number and a hexagonal number. He also raised many open conjectures on congruences and posed over

neg: Waring's problem these lower bounds were in fact exact values. Lagrange's four-square theorem of 1770 states that every natural number is the sum of at most four squares. Since three squares are not enough, this theorem establishes formula_9. Lagrange's four-square theorem was conjectured in Bachet's 1621 edition of Diophantus's Arithmetica; Fermat claimed to have a proof, but did not publish it. Over the years various bounds were established, using increasingly sophisticated and complex proof techniques. For example, Liouville showed that formula_10 is at most 53. Hardy and Littlewood showed that all sufficiently large numbers are the sum of at most 19 fourth

neg: Arithmetica He also noticed that numbers of the form formula_1 cannot be the sum of two squares. Diophantus also appears to know that every number can be written as the sum of four squares. If he did know this result (in the sense of having proved it as opposed to merely conjectured it), his doing so would be truly remarkable: even Fermat, who stated the result, failed to provide a proof of it and it was not settled until Joseph Louis Lagrange proved it using results due to Leonhard Euler. "Arithmetica" was originally written in thirteen books, but the Greek manuscripts

neg: Euler's factorization method essentially the same as with finding differences of squares in Fermat's factorization method. The great disadvantage of Euler's factorization method is that it cannot be applied to factoring an integer with any prime factor of the form 4"k" + 3 occurring to an odd power in its prime factorization, as such a number can never be the sum of two squares. Even odd composite numbers of the form 4"k" + 1 are often the product of two primes of the form 4"k" + 3 (e.g. 3053 = 43 × 71) and again cannot be factored by Euler's method. This restricted

neg: Explained sum of squares greater the ESS, the better the estimated model performs. If formula_1 and formula_2 are the estimated coefficients, then is the "i" predicted value of the response variable. The ESS is the sum of the squares of the differences of the predicted values and the mean value of the response variable: In some cases (see below): total sum of squares = explained sum of squares + residual sum of squares. The following equality, stating that the total sum of squares equals the residual sum of squares plus the explained sum of squares, is generally true in simple linear regression: Square both

neg: Octahedral number tetrahedra each having two adjacent original faces (or alternatively, based on the fact that each square pyramidal number is the sum of two tetrahedral numbers): If two tetrahedra are attached to opposite faces of an octahedron, the result is a rhombohedron. The number of close-packed spheres in the rhombohedron is a cube, justifying the equation The difference between two consecutive octahedral numbers is a centered square number: Therefore, an octahedral number also represents the number of points in a square pyramid formed by stacking centered squares; for this reason, in his book "Arithmeticorum libri duo" (1575), Francesco Maurolico called these

neg: Algebraic number theory to Diophantine equations, named after the 3rd-century Alexandrian mathematician, Diophantus, who studied them and developed methods for the solution of some kinds of Diophantine equations. A typical Diophantine problem is to find two integers "x" and "y" such that their sum, and the sum of their squares, equal two given numbers "A" and "B", respectively: Diophantine equations have been studied for thousands of years. For example, the solutions to the quadratic Diophantine equation "x" + "y" = "z" are given by the Pythagorean triples, originally solved by the Babylonians (c. 1800 BC). Solutions to linear Diophantine equations, such as 26"x"

neg: Cannonball problem Cannonball problem In the mathematics of figurate numbers, the cannonball problem asks which numbers are both square and square pyramidal. The problem can be stated as: given a square arrangement of cannonballs, for what size squares can these cannonballs also be arranged into a square pyramid. Equivalently, which squares can be represented as the sum of consecutive squares, starting from 1? When cannonballs are stacked within a square frame, the number of balls is a square pyramidal number; Thomas Harriot gave a formula for this number around 1587, answering a question posed to him by Sir Walter Raleigh on their

neg: Spectrum (functional analysis) contains all its eigenvalues, but is not limited to them. For example, consider the Hilbert space formula_24, that consists of all bi-infinite sequences of real numbers that have a finite sum of squares formula_26. The bilateral shift operator formula_2 simply displaces every element of the sequence by one position; namely if formula_28 then formula_29 for every integer formula_30. The eigenvalue equation formula_31 has no solution in this space, since it implies that all the values formula_32 have the same absolute value (if formula_33) or are a geometric progression (if formula_34); either way, the sum of their squares would not be

neg: Parshvanatha temple, Khajuraho The sum of the numbers in every horizontal row, every vertical column and the two diagonal rows is 34. The sum of the numbers in the smaller 2x2 squares at the four corners, as well as those in the central 2x2 square, is also 34. The magic square is pandiagonal, aka diabolic: the numbers in its broken diagonals also sum up to 34. For example, 7 + (6 + 10 + 11) = 34. Parshvanatha temple, Khajuraho Parshvanatha temple (IAST: Pārśvanātha Mandir) is a 10th-century Jain temple at Khajuraho in Madhya Pradesh, India. It is now dedicated to Parshvanatha, although

neg: Most-perfect magic square second property above implies that each pair of the integers with the same background colour in the 4×4 square below have the same sum, and hence any 2 such pairs sum to the magic constant. Most-perfect magic square A most-perfect magic square of doubly even order "n" = 4"k" is a pan-diagonal magic square containing the numbers 1 to "n" with three additional properties: Specific examples of most-perfect magic squares that begin with the 2015 date demonstrate how theory and computer science are able to define this group of magic squares. The 12x12 square below was found by making all

neg: Lagrange's four-square theorem Lagrange's four-square theorem Lagrange's four-square theorem, also known as Bachet's conjecture, states that every natural number can be represented as the sum of four integer squares. where the four numbers formula_2 are integers. For illustration, 3, 31 and 310 can be represented as the sum of four squares as follows: This theorem was proven by Joseph Louis Lagrange in 1770. From examples given in the "Arithmetica" it is clear that Diophantus was aware of the theorem. This book was translated in 1621 into Latin by Bachet (Claude Gaspard Bachet de Méziriac), who stated the theorem in the notes of his

neg: Pythagorean prime representation can be interpreted geometrically: the Pythagorean primes are exactly the odd prime numbers "p" such that there exists a right triangle, with integer legs, whose hypotenuse has length . They are also exactly the prime numbers "p" such that there exists a right triangle with integer sides whose hypotenuse has length "p". For, if the triangle with legs "x" and "y" has hypotenuse length (with "x" > "y"), then the triangle with legs "x" − "y" and 2"xy" has hypotenuse length "p". Another way to understand this representation as a sum of two squares involves Gaussian integers, the complex

neg: Two New Sciences a contradiction: Yet at the outset we said there are many more numbers than squares, since the larger portion of them are not squares. Not only so, but the proportionate number of squares diminishes as we pass to larger numbers. He resolves the contradiction by denying the possibility of comparing infinite numbers (and of comparing infinite and finite numbers): We can only infer that the totality of all numbers is infinite, that the number of squares is infinite, and that the number of their roots is infinite; neither is the number of squares less than the totality of all numbers,

neg: Fermat's theorem on sums of two squares he announced the following two results for odd primes formula_4: He also wrote: In other words, if "p, q" are of the form 20"k" + 3 or 20"k" + 7, then "pq" = "x" + 5"y". Euler later extended this to the conjecture that Both Fermat's assertion and Euler's conjecture were established by Lagrange. A generalization of Fermat's theorem, the sum of two squares theorem, characterizes the integers (not necessarily prime) that can be expressed as the sum of two squares. They are exactly the integers in which each prime that is congruent to 3 mod 4 appears with an

neg: Fermat's right triangle theorem that the numbers to be squared are the difference of legs, hypotenuse, and sum of legs of a Pythagorean triangle, and that the congruum is four times the area of the same triangle. In his later work on the congruum problem, published in "The Book of Squares", Fibonacci observed that it is impossible for a congruum to be a square number itself, but did not present a satisfactory proof of this fact. If three squares formula_4, formula_5, and formula_6 could form an arithmetic progression whose congruum was also a square formula_7, then these numbers would satisfy the Diophantine equations That

neg: Magic square two conditions ensures that the resulting square is a semi-magic square; and such Greek and Latin squares are said to be "mutually orthogonal" to each other. To construct a magic square, we should also ensure that the diagonals sum to magic constant. For this, we have a third condition: The mutually orthogonal Greek and Latin squares that satisfy the first part of the third condition (that all letters appear in both the diagonals) are said to be "mutually orthogonal doubly diagonal Graeco-Latin squares". Odd squares: For the odd square, since "α", "β", and "γ" are in arithmetic progression, their sum

neg: Happy number Happy number A happy number is defined by the following process: Starting with any positive integer, replace the number by the sum of the squares of its digits in base-ten, and repeat the process until the number either equals 1 (where it will stay), or it loops endlessly in a cycle that does not include 1. Those numbers for which this process ends in 1 are happy numbers, while those that do not end in 1 are unhappy numbers (or sad numbers). The origin of happy numbers is not clear. Happy numbers were brought to the attention of Reg Allenby

neg: Happy number the associated sequence is The 143 happy numbers up to 1000 are: The happiness of a number is unaffected by rearranging the digits and by inserting or removing any number of zeros anywhere in the number. The distinct combinations of digits that form happy numbers below 1000 are (the rest are just rearrangements and/or insertions of zero digits): Numbers that are happy follow a sequence that ends in 1. All non-happy numbers follow sequences that reach the cycle To see this fact, first note that if "n" has "m" digits, then the sum of the squares of its digits is

neg: Central binomial coefficient the sum of the squares of the elements in row "n" of Pascal's triangle. The closely related Catalan numbers "C" are given by: A slight generalization of central binomial coefficients is to take them as formula_15, with appropriate real numbers "n", where formula_16 is the gamma function and formula_17 is the beta function. The powers of two that divide the central binomial coefficients are given by Gould's sequence, whose "n"th element is the number of odd integers in row "n" of Pascal's triangle. Central binomial coefficient In mathematics the "n"th central binomial coefficient is the particular binomial coefficient They are

neg: Sum of squares function Sum of squares function The sum of squares function is an arithmetic function that gives the number of representations for a given positive integer as the sum of squares, where representations that differ only in the order of the summands or in the signs of the square roots are counted as different, and is denoted by . The function is defined as where |.| denotes the cardinality of the set. In other words, is the number of ways can be written as a sum of squares. For example, formula_2, since formula_3, where every sum has 2 sign combinations, and also

neg: Proofs of Fermat's theorem on sums of two squares formula_196. The area of the fundamental parallelogram (in this case a rectangle) of the lattice is formula_82. The area of the open disk, formula_198, of radius formula_199 centered around the origin is formula_200. Furthermore, formula_198 is convex and symmetrical about the origin. Therefore, by Minkowski's theorem there exists a nonzero vector formula_196 such that formula_203. Both formula_204 and formula_205 so formula_206. Hence formula_82 is the sum of the squares of the components of formula_208. Let formula_209 be prime, let formula_210 denote the natural numbers (with or without zero), and consider the finite set formula_211 of triples of numbers. Then formula_212

neg: Happy number numbers. The definition can be extended to other bases. To represent numbers in other bases, we may use a subscript to the right to indicate the base. For instance, formula_15 represents the number 4, and Then, it is easy to see that there are happy numbers in every base. For instance, the numbers are all happy, for any base "b". By a similar argument to the one above for decimal happy numbers, unhappy numbers in base "b" lead to cycles of numbers less than formula_18. If formula_19, then the sum of the squares of the base-"b" digits of "n" is

neg: Magic square and right way up: Sometimes the rules for magic squares are relaxed, so that only the rows and columns but not necessarily the diagonals sum to the magic constant (this is usually called a semimagic square). In heterosquares and antimagic squares, the 2"n" + 2 sums must all be "different". Instead of "adding" the numbers in each row, column and diagonal, one can apply some other operation. For example, a multiplicative magic square has a constant "product" of numbers. A multiplicative magic square can be derived from an additive magic square by raising 2 (or any other integer) to the

neg: Lack-of-fit sum of squares of squares due to "pure" error is the sum of squares of the differences between each observed "y"-value and the average of all "y"-values corresponding to the same "x"-value. The sum of squares due to lack of fit is the "weighted" sum of squares of differences between each average of "y"-values corresponding to the same "x"-value and the corresponding fitted "y"-value, the weight in each case being simply the number of observed "y"-values for that "x"-value. Because it is a property of least squares regression that the vector whose components are "pure errors" and the vector of lack-of-fit components are

neg: Happy number search then shows that every number in the interval [1, 99] either is happy or goes to the above cycle. The above work produces the interesting result that no positive integer other than 1 is the sum of the squares of its own digits, since any such number would be a fixed point of the described process. There are infinitely many happy numbers and infinitely many unhappy numbers. Consider the following proof: Indeed, the "happiness" of a number is preserved by removing or inserting zeroes at will, since they do not contribute to the cross sum. See the above proof,

neg: Goldbach's conjecture accepted by the mathematical community. Although Goldbach's conjecture implies that every positive integer greater than one can be written as a sum of at most three primes, it is not always possible to find such a sum using a greedy algorithm that uses the largest possible prime at each step. The Pillai sequence tracks the numbers requiring the largest number of primes in their greedy representations. One can consider similar problems in which primes are replaced by other particular sets of numbers, such as the squares. Goldbach's conjecture Goldbach's conjecture is one of the oldest and best-known unsolved problems in

neg: Legendre's three-square theorem integer which is neither of the form 8"n" + 7, nor of the form 4"n", is the sum of three squares, but did not provide a satisfactory proof. In 1796 Gauss proved his Eureka theorem that every positive integer "n" is the sum of 3 triangular numbers; this is equivalent to the fact that 8"n" + 3 is a sum of three squares. In 1797 or 1798 A.-M. Legendre obtained the first proof of his 3 square theorem. In 1813, A. L. Cauchy noted that Legendre's theorem is equivalent to the statement in the introduction above. Previously, in 1801, C.

neg: Sum of squares function and formula_13 are the prime factors of the form formula_14 gives another formula The number of ways to represent as the sum of four squares was due to Carl Gustav Jakob Jacobi and it is eight times the sum of all its divisors which are not divisible by 4, i.e. Jacobi also found an explicit formula for the case : The generating series that gives the coefficients of the general form is based in terms of Jacobi theta function: where Sum of squares function The sum of squares function is an arithmetic function that gives the number of representations for

neg: Erdős–Turán conjecture on additive bases basis of order 1, since every natural number is trivially a sum of at most one natural number. It is a non-trivial theorem of Lagrange (Lagrange's four-square theorem) that the set of positive square numbers is an additive basis of order 4. Another highly non-trivial and celebrated result along these lines is Vinogradov's theorem. One is naturally inclined to ask whether these results are optimal. It turns out that Lagrange's four-square theorem cannot be improved, as there are infinitely many positive integers which are not the sum of three squares. This is because no positive integer which is the sum

neg: Sum of squares function formula_4, since formula_5 with 4 sign combinations. On the other hand is formula_6, because there exists no way to represent 3 as a sum of two squares. The number of ways to write a natural number as sum of two squares is given by . It is given explicitly by where is the number of divisors of which are congruent with 1 modulo 4 and is the number of divisors of which are congruent with 3 modulo 4. Using sums, the expression can be written as: The prime factorization formula_10, where formula_11 are the prime factors of the form formula_12

neg: Fermat's theorem on sums of two squares version of the statement (in which he also gave the number of possible expressions of the powers of "p" as a sum of two squares) in a letter to Marin Mersenne dated December 25, 1640: for this reason this version of the theorem is sometimes called "Fermat's Christmas theorem." Since the Diophantus identity implies that the product of two integers each of which can be written as the sum of two squares is itself expressible as the sum of two squares, by applying Fermat's theorem to the prime factorization of any positive integer "n", we see that if all the

neg: Sum-of-squares optimization SOS polynomials and positive-semidefinite matrices. Sum-of-squares optimization A sum-of-squares optimization program is an optimization problem with a linear cost function and a particular type of constraint on the decision variables. These constraints are of the form that when the decision variables are used as coefficients in certain polynomials, those polynomials should have the polynomial SOS property. When fixing the maximum degree of the polynomials involved, sum-of-squares optimization is also known as the Lasserre hierarchy of relaxations in semidefinite programming. Sum-of-squares optimization techniques have been successfully applied by researchers in the control engineering field, in special in the search for polynomial

neg: Sum-of-squares optimization Sum-of-squares optimization A sum-of-squares optimization program is an optimization problem with a linear cost function and a particular type of constraint on the decision variables. These constraints are of the form that when the decision variables are used as coefficients in certain polynomials, those polynomials should have the polynomial SOS property. When fixing the maximum degree of the polynomials involved, sum-of-squares optimization is also known as the Lasserre hierarchy of relaxations in semidefinite programming. Sum-of-squares optimization techniques have been successfully applied by researchers in the control engineering field, in special in the search for polynomial Lyapunov functions for dynamical systems

neg: Antimagic square antimagic square on the right, the rows, columns and diagonals add up to numbers between 59-70. Antimagic squares form a subset of heterosquares which simply have each row, column and diagonal sum different. They contrast with magic squares where each sum is the same. A sparse antimagic square (SAM) is a square matrix of size "n" by "n" of nonnegative integers whose nonzero entries are the consecutive integers formula_1 for some formula_2, and whose row-sums and column-sums constitute a set of consecutive integers. If the diagonals are included in the set of consecutive integers, the array is known as a

neg: Albert Girard Albert Girard Albert Girard () (1595 in Saint-Mihiel, France − 8 December 1632 in Leiden) was a French-born mathematician. He studied at the University of Leiden. He "had early thoughts on the fundamental theorem of algebra" and gave the inductive definition for the Fibonacci numbers. He was the first to use the abbreviations 'sin', 'cos' and 'tan' for the trigonometric functions in a treatise. Girard was the first to state, in 1625, that each prime of the form 1 mod 4 is the sum of two squares. (See Fermat's theorem on sums of two squares.) It was said that he

neg: Prime number is a consequence of Fermat's theorem on sums of two squares, which states that an odd prime formula_20 is expressible as the sum of two squares, formula_187, and therefore factorizable as formula_188, exactly when formula_20 is 1 mod 4. Not every ring is a unique factorization domain. For instance, in the ring of numbers formula_190 (for integers formula_34 and formula_35) the number formula_193 has two factorizations formula_194, where neither of the four factors can be reduced any further, so it does not have a unique factorization. In order to extend unique factorization to a larger class of rings, the notion

neg: Euler's factorization method applicability has made Euler's factorization method disfavoured for computer factoring algorithms, since any user attempting to factor a random integer is unlikely to know whether Euler's method can actually be applied to the integer in question. It is only relatively recently that there have been attempts to develop Euler's method into computer algorithms for use on specialised numbers where it is known Euler's method can be applied. The Brahmagupta–Fibonacci identity states that the product of two sums of two squares is a sum of two squares. Euler's method relies on this theorem but it can be viewed as the converse,

neg: Pandiagonal magic cube Pandiagonal magic cube In recreational mathematics, a pandiagonal magic cube is a magic cube with the additional property that all broken diagonals (parallel to exactly two of the three coordinate axes) have the same sum as each other. Pandiagonal magic cubes are extensions of diagonal magic cubes (in which only the unbroken diagonals need to have the same sum as the rows of the cube) and generalize pandiagonal magic squares to three dimensions. In a pandiagonal magic cube, all 3m planar arrays must be panmagic squares. The 6 oblique squares are always magic. Several of them may be panmagic squares.

neg: Multiplication algorithm squares with the remainder discarded for the digits 0 through 18; this allows for the multiplication of numbers up to . If, for example, you wanted to multiply 9 by 3, you observe that the sum and difference are 12 and 6 respectively. Looking both those values up on the table yields 36 and 9, the difference of which is 27, which is the product of 9 and 3. Antoine Voisin published a table of quarter squares from 1 to 1000 in 1817 as an aid in multiplication. A larger table of quarter squares from 1 to 100000 was published

neg: Explained sum of squares underlying errors. The ordinary least squares estimator for formula_16 is The residual vector formula_18 is formula_19, so the residual sum of squares formula_20 is, after simplification, Denote as formula_22 the constant vector all of whose elements are the sample mean formula_23 of the dependent variable values in the vector "y". Then the total sum of squares is The explained sum of squares, defined as the sum of squared deviations of the predicted values from the observed mean of "y", is Using formula_26 in this, and simplifying to obtain formula_27, gives the result that "TSS" = "ESS" + "RSS" if and

neg: Magic square power of each element, because the logarithm of the product of 2 numbers is the sum of logarithm of each. Alternatively, if any 3 numbers in a line are 2, 2 and 2, their product is 2, which is constant if "a"+"b"+"c" is constant, as they would be if "a", "b" and "c" were taken from ordinary (additive) magic square. For example, the original Lo-Shu magic square becomes: Other examples of multiplicative magic squares include: Still using Ali Skalli's non iterative method, it is possible to produce an infinity of multiplicative magic squares of complex numbers belonging to formula_17 set.

neg: Magic square accomplished in the late 20th century. In regard to magic sum, the problem of magic squares only requires the sum of each row, column and diagonal to be equal, it does not require the sum to be a particular value. Thus, although magic squares may contain negative integers, they are just variations by adding or multiplying a negative number to every positive integer in the original square. Magic squares composed of integers formula_2 are also called "normal magic squares", in the sense that there are "non-normal magic squares" whose integers are not restricted to formula_2. However, in some places, ""magic

neg: Erdős–Turán conjecture on additive bases those positive integers formula_4 that passed the arithmetic obstruction. Examining Vinogradov's theorem quickly reveals that the primes are also very inefficient at representing positive integers as the sum of four primes, for instance. This begets the question: suppose that formula_2, unlike the squares or the prime numbers, is very efficient at representing positive integers as a sum of formula_3 elements of formula_2. How efficient can it be? The best possibility is that we can find a positive integer formula_3 and a set formula_2 such that every positive integer formula_4 is the sum of at most formula_3 elements of formula_2 in

neg: Kakuro sum could have those values. For instance, if two 4-in-two clues cross with a longer sum, then the 1 and 3 in the solution must be in those two squares and those digits cannot be used elsewhere in that sum. When solving sums which have a limited number of solution sets then that can lead to useful clues. For instance, a 30-in-seven sum only has two solution sets: {1,2,3,4,5,6,9} and {1,2,3,4,5,7,8}. If one of the squares in that sum can only take on the values of {8,9} (if the crossing clue is a 17-in-two sum, for example) then that not

neg: Unistochastic matrix the set of unistochastic matrices. Unistochastic matrix In mathematics, a unistochastic matrix (also called "unitary-stochastic") is a doubly stochastic matrix whose entries are the squares of the absolute values of the entries of some unitary matrix. A square matrix "B" of size "n" is doubly stochastic (or "bistochastic") if all its entries are non-negative real numbers and each of its rows and columns sum to 1. It is unistochastic if there exists a unitary matrix "U" such that This definition is analogous to that for an orthostochastic matrix, which is a doubly stochastic matrix whose entries are the squares of

neg: Unistochastic matrix Unistochastic matrix In mathematics, a unistochastic matrix (also called "unitary-stochastic") is a doubly stochastic matrix whose entries are the squares of the absolute values of the entries of some unitary matrix. A square matrix "B" of size "n" is doubly stochastic (or "bistochastic") if all its entries are non-negative real numbers and each of its rows and columns sum to 1. It is unistochastic if there exists a unitary matrix "U" such that This definition is analogous to that for an orthostochastic matrix, which is a doubly stochastic matrix whose entries are the squares of the entries in some orthogonal

neg: Mixed-design analysis of variance term for the first and second factors, and the error term. The main difference between the sum of squares of the within-subject factors and between-subject factors is that within-subject factors have an interaction factor. More specifically, the total sum of squares in a regular one-way ANOVA would consist of two parts: variance due to treatment or condition (SS) and variance due to error (SS). Normally the SS is a measurement of variance. In a mixed-design, you are taking repeated measures from the same participants and therefore the sum of squares can be broken down even further into three components: SS

neg: Lack-of-fit sum of squares Lack-of-fit sum of squares In statistics, a sum of squares due to lack of fit, or more tersely a lack-of-fit sum of squares, is one of the components of a partition of the sum of squares of residuals in an analysis of variance, used in the numerator in an F-test of the null hypothesis that says that a proposed model fits well. The other component is the pure-error sum of squares. The pure-error sum of squares is the sum of squared deviations of each value of the dependent variable from the average value over all observations sharing its independent variable

neg: Composition algebra sums of squares was noted by several early authors. Diophantus was aware of the identity involving the sum of two squares, now called the Brahmagupta–Fibonacci identity, which is also articulated as a property of Euclidean norms of complex numbers when multiplied. Leonhard Euler discussed the four-square identity in 1748, and it led W. R. Hamilton to construct his four-dimensional algebra of quaternions. In 1848 tessarines were described giving first light to bicomplex numbers. About 1818 Danish scholar Ferdinand Degen displayed the Degen's eight-square identity, which was later connected with norms of elements of the octonion algebra: In 1919 Leonard Dickson

neg: Multiplicative number theory estimates the average order of the divisor function "d(n)" and Gauss's circle problem that estimates the average order of the number of representations of a number as a sum of two squares are also classical problems, and again the focus is on improving the error estimates. The distribution of primes numbers among residue classes modulo an integer is an area of active research. Dirichlet's theorem on primes in arithmetic progressions shows that there are an infinity of primes in each co-prime residue class, and the prime number theorem for arithmetic progressions shows that the primes are asymptotically equidistributed among the

neg: Automedian triangle difference between twice a square and a square. They are also the sum and difference of the legs of a primitive Pythagorean triple. This constrains "b" and "c" to be divisible only by primes congruent to ±1 (mod 8). Consequently, "b" and "c" must be congruent to ±1 (mod 8). The study of integer squares in arithmetic progression has a long history stretching back to Diophantus and Fibonacci; it is closely connected with congrua, which are the numbers that can be the differences of the squares in such a progression. However, the connection between this problem and automedian triangles is

neg: Happy number 1, and one cycle: The status in base 16 is similar to base 10. A variation to the happy numbers problem is to find the sum of the cubes of the digits rather than the sum of the squares of the digits. For example, working in base 10, 1579 is happy, since: In the same way that when summing the squares of the digits (and working in base 10) each number above 243(=3*81) produces a number that is strictly smaller, when summing the cubes of the digits each number above 2916(=4*729) produces a number that is strictly smaller. By conducting

neg: Fraction of variance unexplained and the sample variance of the dependent variable. "SS" (the sum of squared predictions errors, equivalently the residual sum of squares), "SS" (the total sum of squares), and "SS" (the sum of squares of the regression, equivalently the explained sum of squares) are given by Alternatively, the fraction of variance unexplained can be defined as follows: where MSE("f") is the mean squared error of the regression function "ƒ". It is useful to consider the second definition to understand FVU. When trying to predict "Y", the most naïve regression function that we can think of is the constant function predicting the