Gauss' binomial coefficient congruence, and its extension ("suggested by Beukers"), proved by Chowla-Dwork-Evans 

Gauss' binomial coefficient congruence theorem. For every prime    (mod 4) we have: 

(mod )  

where '' is chosen according to the modified Fermat two-square theorem above. 

Remark #1. The object on the left-hand side is clearly a binomial coefficient. 

Remark #2. Gauss gave (and proved) this remarkable result in his 1828 paper as a result of studying the question: 

For primes , how many solutions does the congruence have?

 

Remark #3. I recommend my Monthly paper with Karl Dilcher (paper number 4 at https://johnbcosgrave.com/publications) for an alternative way of viewing (I mean by way of discovery) Gauss' theorem. 

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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_sign(p)^2);             BC||p := ((p-1)/2)!/((p-1)/4)!^2:           od: print(``); print(array([ [ 'p', ``, ``, 'p = a^2 + b^2', ``, '({p-1}/2)!/({p-1}/4)!^2', ``, ``,    'BC_red', ``, ``, '2*a'], seq( [ p, ``, ``, [a||p, b||p], ``,  BC||p, ``, ``, mods(BC||p, p), ``, ``, 2*a||p ], p = L ) ] ) ): print(``); lprint(`The 2nd last column is the least absolute residue of the central column,`); print(``); lprint(`while the final column is the least absolute residue of 2 times the signed a.`);

 

 

 

 

 

 

`The 2nd last column is the least absolute residue of the central column,`
`while the final column is the least absolute residue of 2 times the signed a.`

 

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The Chowla-Dwork-Evans Theorem (1986). For every prime    (mod 4) we have: 

(mod )  

where is the (already encountered) base-2 Fermat quotient.