A couple of Number Theory questions

[squire636]

1. Find all solutions x (with 0 ≤ x ≤ 96) to the congruence 13x^385 + 73x^304 + x^290 + 10x^193 + 24x^112 + 70x + 76 ≡ 0 (mod 97)

I was able to reduce, using Fermat's Little Theorem, to get 97x^16 + x^2 + 93x + 76 ≡ 0 (mod 97), but I don't know how to proceed from there. Is there another trick I can use?2. http://imgur.com/a/DUyHC

RR denotes two adjacent quadratic residues, while NN denotes two adjacent quadratic non-residues. RN is a residue followed by a non-residue, and NR is a non-residue followed by a residue. I tried to solve the problems by adding and subtracting the four expressions given in part b but I haven't made any progress.Thanks for the help!

 

[SammyS]

1. Find all solutions x (with 0 ≤ x ≤ 96) to the congruence 13x^385 + 73x^304 + x^290 + 10x^193 + 24x^112 + 70x + 76 ≡ 0 (mod 97)

I was able to reduce, using Fermat's Little Theorem, to get 97x^16 + x^2 + 93x + 76 ≡ 0 (mod 97), but I don't know how to proceed from there. Is there another trick I can use?

2. http://imgur.com/a/DUyHC

RR denotes two adjacent quadratic residues, while NN denotes two adjacent quadratic non-residues. RN is a residue followed by a non-residue, and NR is a non-residue followed by a residue. I tried to solve the problems by adding and subtracting the four expressions given in part b but I haven't made any progress.

Thanks for the help!

One quick question: What is 97 mod 97 ?

 

[squire636]

One quick response: UGHHHH sometimes I'm an idiot. Thanks!

 

[SammyS]

One quick response: UGHHHH sometimes I'm an idiot. Thanks!

Did you find the solution ?

 

[squire636]

I sure did, I found two solutions just by using the quadratic formula on the equation that remains after the x^16 term goes to zero. Then I just had to make sure that they're between 0 and 97. My solutions aren't integers, which is sort of frustrating, but oh well.

Any advice on the other problem?

 

[SammyS]

I sure did, I found two solutions just by using the quadratic formula on the equation that remains after the x^16 term goes to zero. Then I just had to make sure that they're between 0 and 97. My solutions aren't integers, which is sort of frustrating, but oh well.

Any advice on the other problem?

Well, there are two integer solutions.

What numbers are congruent to 93 and/or 76 (mod 97) ?

 

[squire636]

Oh good call, thanks so much. I got integer solutions once I changed the x and the constant term to congruent values.