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