Find all seventhpower residues modulo 29SolutionAns You coul

Find all seventh-power residues modulo 29.

Solution

Ans-

You could check 07,17,27,…,14707,17,27,…,147 mod 2929. Then

{07mod29,17mod29,27mod29,…,147mod29,{07mod29,17mod29,27mod29,…,147mod29,

147mod29,…,27mod29,17mod29}147mod29,…,27mod29,17mod29}

would be your answer. But this could be a bit tedious. Here is another solution:

aa is a 77\'th power mod 2929 if and only if ax7(mod29)ax7(mod29) for some xZxZ.

Clearly 00 is a 77\'th power mod 2929. Let 29a29a. Then if ax7(mod29)ax7(mod29) for some xZxZ, then a4x281(mod29)a4x281(mod29) by Fermat\'s Little Theorem.

29a41=(a1)(a+1)(a2+1)29a41=(a1)(a+1)(a2+1)

By Euclid\'s Lemma this is equivalent to either 29a129a1 or 29a+129a+1 or 29a2+129a2+1.

If a21(mod29)a21(mod29), then a2144122(mod29)a2144122(mod29), so

29a2122=(a+12)(a12)29a2122=(a+12)(a12). By Euclid\'s Lemma either 29a+1229a+12 or 29a1229a12.

Therefore {0,1,1,12,12}{0,1,1,12,12} is your answer, or i.e. {0,1,12,17,28}{0,1,12,17,28}.