I am working on some homework that I already handed in, but I cant get one of the problems. The fourth problem on the HW was to prove the forms of (-1/p), (2/p), (3/p), (-5/p), and (7/p).(adsbygoogle = window.adsbygoogle || []).push({});

I did this for -1 and 2 using the quadratic residues and generalizing a form for them. for 3 and 7 i used QRL, since they are both -1 mod 4, can i use QRL for the proof of -5 too? i know i got at least 80% on this problem, and thats a B+, so i should be fine on this problem. could someone please guide me on the first steps of this proof so that i can understand it? 3 and 7 were pretty easy, but im not sure i got 7 right. most of it was in the book by David Burton that we use. BTW, Im a sophomore in math, so this class is really hard for me. thats why im coming here for more understanding, that and my profs office hours are short and i use them for linear algebra.

for 3, i showed p congruent to 1 mod 4 for 4|p-1 and congruent to 1 mod 3 for 3|p-1, so 12|p-1, the forms of this p congruent to 3 mod 4 are 3 mod 12, 7 mod 12, 11 mod 12, and p congruent to 2 mod 3, if p congruent to 2 mod 12, 5 mod 12, 8 mod 12, 11 mod 12. the common solutions are p congruent to 1 and 11 mod 12, so its +- 1 mod 12, (3/p)=1, and since 8 is 0 mod 4, toss it, 3 and 9 are 0 mod 3, toss em, so 5,7 yield +- 5 mod 12, (3/p)=-1

can someone lead me through this for -5 now?

sorry for type settting, it wasnt really that necessary for this problem, and im in a lab where i dont have much time left. sorry for long paragraphs too!

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# Legendre symbol proof for (-5/p)

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