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Pierre de Fermat gave a criterion for numbers of the form 8a + 1 and 8a + 3 to be sums of a square plus twice another square, but did not provide a proof. [1] N. Beguelin noticed in 1774 [2] that every positive integer which is neither of the form 8n + 7, nor of the form 4n, is the sum of three squares, but did not provide a satisfactory proof. [3]
Legendre's three-square theorem states which numbers can be expressed as the sum of three squares; Jacobi's four-square theorem gives the number of ways that a number can be represented as the sum of four squares. For the number of representations of a positive integer as a sum of squares of k integers, see Sum of squares function.
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 4 k (8m + 7).
Additionally, multiplying a congruum by a square number produces another congruum, whose progression of squares is multiplied by the same factor. All solutions arise in one of these two ways. [ 1 ] For instance, the congruum 96 can be constructed by these formulas with m = 3 {\displaystyle m=3} and n = 1 {\displaystyle n=1} , while the congruum ...
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The folks at Minyanville did a comprehensive calculation and came up with the following chart. The numbers are based on a $50 a square game, with a $625 payout for the 1st and 3rd quarters, a ...
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The number of representations of a natural number n as the sum of four squares of integers is denoted by r 4 (n). Jacobi's four-square theorem states that this is eight times the sum of the divisors of n if n is odd and 24 times the sum of the odd divisors of n if n is even (see divisor function), i.e.