Fermat's two-square theorem (with the 'signed' element incorporated into its statement) 

 

 

Fermat's (beautiful) two square theorem (in modified form, for the purposes of the upcoming Gauss binomial coefficient congruence).

For every prime p = 1  (mod 4) there are unique integers, odd (which may involve a sign change) and (positive) b such that 

 

p = `+`(`*`(`^`(a, 2)), `*`(`^`(b, 2))) 

 

 

Examples to illustrate. 

 

 

 

 

 

Here I exhibit these representation for all such primes up to 200: 

 

 

> L := []:

for p from 5 by 4 to 200 do if isprime(p) then

L := [op(L), p] fi od:

for p in L do

             a||p := a_sign(p): ### see the Procedures section

             b||p := sqrt(p - a||p^2):

     od:

print(``); print(array([

[  'p', ``, ``, 'a', ``, ``, 'b'  ],

 seq( [p, ``, ``, a||p, ``, ``, b||p ],

 p = L ) ] ) ): print(``);

lprint(`This shows the UNIQUE a = 1 (mod 4) and (positive) b in the modified Fermat two-square theorem.`);
 

 

 

 

array( 1 .. 22, 1 .. 7, [( 18, 5 ) = (``), ( 7, 3 ) = (``), ( 19, 4 ) = (13), ( 6, 4 ) = (1), ( 16, 7 ) = (4), ( 5, 5 ) = (``), ( 17, 6 ) = (``), ( 4, 6 ) = (``), ( 22, 1 ) = (197), ( 10, 6 ) = (``), ...
`This shows the UNIQUE a = 1 (mod 4) and (positive) b in the modified Fermat two-square theorem.`
 

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