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 April 24th, 2019, 06:17 AM #1 Senior Member   Joined: Jan 2016 From: Blackpool Posts: 104 Thanks: 2 prime number question Find the smallest number n for which 11 divides 20n^(2)-1. For this question, I have made it so that 20n^2 is congruent to -1mod11; the quadratic residues of 11 are 1 goes to 1, 2 goes to 4, 3 goes to 9, 4 goes to 5, 5 goes to 3. But none of these contain the number -1, so -1 does not belong in the set of squares, so how am I supposed to work out this question? I must have done something wrong in my understanding, so could anyone explain to me what I have done wrong? Thanks! Last edited by skipjack; April 24th, 2019 at 10:11 AM. April 24th, 2019, 07:37 AM #2 Senior Member   Joined: Dec 2015 From: somewhere Posts: 605 Thanks: 88 Try $\displaystyle 20n^2 -1 =11n^2 +9n^2 -1$ , now $\displaystyle 11$ divides $\displaystyle 9n^2 -1 =(3n-1)(3n+1)$. Thanks from Jaket1 Last edited by idontknow; April 24th, 2019 at 07:43 AM. April 24th, 2019, 07:57 AM #3 Senior Member   Joined: Mar 2019 From: iran Posts: 318 Thanks: 14 20 n^2 = 1 (mod 11) n = 4 20 × 4^2 = 1 (mod 11) Thanks from idontknow and Jaket1 April 24th, 2019, 10:07 AM #4 Senior Member   Joined: Dec 2015 From: somewhere Posts: 605 Thanks: 88 You can solve the problem in many ways but this one doesn’t need complicated theorems of twin primes. 3n-1 and 3n+1 must be both prime and $\displaystyle n=2k$ which leads to the known form of twin primes $\displaystyle (6k-1,6k+1)$. A weak shortcut just to prove n=4 : $\displaystyle 3n-1 \geq 12$ or $\displaystyle n\geq 13/3 > 12/3=4$. Thanks from Jaket1 Last edited by idontknow; April 24th, 2019 at 10:22 AM. Tags number, prime, question Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post CSandMathsGuy Number Theory 3 August 16th, 2018 12:14 PM idontknow Number Theory 12 July 18th, 2018 06:19 AM thinhnghiem Math 0 May 15th, 2018 08:07 AM thinhnghiem Number Theory 15 December 18th, 2016 01:31 PM Infinity Number Theory 5 September 25th, 2007 07:02 AM

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