| Lagrange's Four-square Theorem |
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It states that every Positive Integer can be expressed as the sum of four squares of integers. For example, :31 = 52 + 22 + 12 + 12 :310 = 172 + 42 + 22 + 12. More formally, for every positive integer n there exist non-negative integers a,b,c,d such that n Adrien-Marie Legendre improved on the theorem in 1798 by stating that a positive integer can be expressed as the sum of three squares Iff it is not of the form 4''k''(8''m'' + 7). His proof was incomplete, leaving a gap which was later filled by Carl Friedrich Gauss . In 1834 , Carl Gustav Jakob Jacobi found an exact formula for the total number of ways a given positive integer ''n'' can be represented as the sum of four squares. This number is eight times the sum of the Divisor s of ''n'' if ''n'' is odd and 24 times the sum of the odd divisors of ''n'' if ''n'' is even. Lagrange's four-square theorem is a special case of the Fermat Polygonal Number Theorem and Waring's Problem . SEE ALSO REFERENCES EXTERNAL LINKS |