A couple of Number Theory questions

Click For Summary

Homework Help Overview

The discussion revolves around solving congruences in number theory, specifically focusing on a polynomial congruence modulo 97 and properties of quadratic residues and non-residues.

Discussion Character

  • Exploratory, Mathematical reasoning, Problem interpretation

Approaches and Questions Raised

  • Participants discuss reducing a complex polynomial congruence using Fermat's Little Theorem and explore potential methods for further simplification. Questions arise regarding alternative techniques and the implications of quadratic residues and non-residues.

Discussion Status

Some participants have made progress in identifying solutions to the polynomial congruence, while others are still seeking clarification on the properties of quadratic residues. There is acknowledgment of both integer and non-integer solutions, indicating a productive exploration of the problem.

Contextual Notes

Participants mention constraints related to the range of solutions (0 ≤ x ≤ 96) and the nature of the solutions being sought (integer vs. non-integer). There is also a reference to the congruence relationships among the constants involved.

squire636
Messages
38
Reaction score
0
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!
 
Last edited:
Physics news on Phys.org
squire636 said:
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 ?
 
One quick response: UGHHHH sometimes I'm an idiot. Thanks!
 
squire636 said:
One quick response: UGHHHH sometimes I'm an idiot. Thanks!
Did you find the solution ?
 
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?
 
squire636 said:
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) ?
 
Last edited:
Oh good call, thanks so much. I got integer solutions once I changed the x and the constant term to congruent values.
 

Similar threads

Graduate How can I avoid large exponents in calculations with small answers?
  • · Replies 5 ·
Replies
5
Views
3K