## sqrt((808^5-627^5)/(808-627)) is prime

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• Off list, Jane Sullivan remarked on integer values of the square root of the homogeneous cyclotomic polynomial Phi(n;a,b) = Phi(n,a/b)*b^eulerphi(n) for
Message 1 of 55 , Dec 22, 2010
Off list, Jane Sullivan remarked on integer values of
the square root of the homogeneous cyclotomic polynomial
Phi(n;a,b) = Phi(n,a/b)*b^eulerphi(n)
for integers (n,a,b) with n > 2 and a > b > 0.

I found 4 cases in which:
1) eulerphi(n) > 2
2) a > b > 0,
3) gcd(a,b) = 1,
4) Phi(n;a,b) is a square.

{nmax=20;amax=10^3;for(n=3,nmax,e=eulerphi(n);
if(e>2,f=polcyclo(n);for(b=1,amax-1,m=b^e;
for(a=b+1,amax,if(gcd(a,b)==1,N=subst(f,x,a/b)*m;
if(issquare(N,&s),print([n,[a,b],s])))))));}

[5, [3, 1], 11]
[5, [808, 627], 1169341]
[10, [11, 8], 101]
[10, [123, 35], 13361]

Can you find [or estimate the probability for] a 5th case?

David
• We have written up (some of) our findings: http://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind1102&L=nmbrthry&P=R752 Kevin and David
Message 55 of 55 , Feb 11, 2011
We have written up (some of) our findings:

http://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind1102&L=nmbrthry&P=R752

Kevin and David
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