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Fermat's 4n+1 Theorem


Fermat's 4n+1 theorem, sometimes called Fermat's two-square theorem or simply "Fermat's theorem," states that a prime number p can be represented in an essentially unique manner (up to the order of addends) in the form x^2+y^2 for integer x and y iff p=1 (mod 4) or p=2 (which is a degenerate case with x=y=1). The theorem was stated by Fermat, but the first published proof was by Euler.

The first few primes p which are 1 or 2 (mod 4) are 2, 5, 13, 17, 29, 37, 41, 53, 61, ... (OEIS A002313) (with the only prime congruent to 2 mod 4 being 2). The numbers (x,y) such that x^2+y^2 equal these primes are (1, 1), (1, 2), (2, 3), (1, 4), (2, 5), (1, 6), ... (OEIS A002331 and A002330).

The theorem can be restated by letting

 Q(x,y)=x^2+y^2,

then all relatively prime solutions (x,y) to the problem of representing Q(x,y)=m for m any integer are achieved by means of successive applications of the genus theorem and composition theorem.


See also

Choquet Theory, Diophantine Equation--2nd Powers, Eisenstein Integer, Euler's 6n+1 Theorem, Fermat's Little Theorem, Sierpiński's Prime Sequence Theorem, Square Number

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References

Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 146-147 and 220-223, 1996.Hardy, G. H. and Wright, E. M. An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, pp. 13 and 219, 1979.Séroul, R. "Prime Number and Sum of Two Squares." §2.11 in Programming for Mathematicians. Berlin: Springer-Verlag, pp. 18-19, 2000.Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, pp. 142-143, 1993.Sloane, N. J. A. Sequences A002313/M1430, A002330/M000462, and A002331/M0096 in "The On-Line Encyclopedia of Integer Sequences."

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Fermat's 4n+1 Theorem

Cite this as:

Weisstein, Eric W. "Fermat's 4n+1 Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Fermats4nPlus1Theorem.html

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