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Let p>5 be a prime. Find all of the possible remainders that p^2 can leave when divided by 30.

 Jun 13, 2019
 #1
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All the primes > 5 when squared mod 30 give a remainder of either 1 or 19.

 Jun 13, 2019
 #2
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Here is a short computer code to check the first 165 prime numbers between 7 and 1,000:
n=7; c=0;m=1000; a=if(isprime(n), goto4, goto6);print(n^2)%30,", ",;c=c+1; n++; if(n >Total = ",c
[19 , 1 , 19 , 19 , 1 , 19 , 1 , 1 , 19 , 1 , 19 , 19 , 19 , 1 , 1 , 19 , 1 , 19 , 1 , 19 , 1 , 19 , 1 , 19 , 19 , 1 , 19 , 19 , 1 , 19 , 1 , 1 , 1 , 19 , 19 , 19 , 19 , 1 , 1 , 1 , 19 , 19 , 1 , 1 , 19 , 19 , 1 , 19 , 1 , 1 , 1 , 19 , 19 , 1 , 1 , 19 , 1 , 19 , 19 , 19 , 1 , 19 , 19 , 1 , 19 , 19 , 1 , 19 , 1 , 19 , 19 , 1 , 19 , 1 , 19 , 1 , 1 , 1 , 1 , 1 , 19 , 1 , 19 , 1 , 19 , 1 , 19 , 19 , 1 , 19 , 1 , 1 , 19 , 1 , 1 , 19 , 1 , 19 , 19 , 19 , 1 , 1 , 19 , 19 , 19 , 1 , 1 , 19 , 19 , 19 , 1 , 1 , 1 , 19 , 19 , 19 , 1 , 1 , 19 , 19 , 19 , 1 , 1 , 1 , 1 , 19 , 19 , 1 , 19 , 1 , 19 , 1 , 1 , 19 , 19 , 19 , 1 , 1 , 1 , 19 , 19 , 1 , 1 , 19 , 19 , 1 , 19 , 19 , 1 , 19 , 19 , 19 , 1 , 1 , 1 , 19 , 1 , 19 , 19 , 19 , 1 , 19 , 19 , 1 , 19]>> Total =  165

 Jun 13, 2019

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