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number > number theory > algebraic number theory > modular arithmetic > Fermat's theorem on sums of two squares
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number > number theory > algebraic number theory > quadratic integer > Fermat's theorem on sums of two squares
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algebra > polynomial > polynomial equation > Diophantine equation > Fermat's theorem on sums of two squares
mathematical analysis > equation > polynomial equation > Diophantine equation > Fermat's theorem on sums of two squares
number > number theory > analytic number theory > additive number theory > Fermat's theorem on sums of two squares

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Fermat's theorem on sums of two squares  

Definición

  • In additive number theory, Fermat's theorem on sums of two squares states that an odd prime p can be expressed as:
    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, 19, 23 and 31 are all congruent to 3 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.
    (Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Fermat%27s_theorem_on_sums_of_two_squares)

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