[seqfan] Conjecture re the specific pair of coprime polynomials

[Alexander Povolotsky]

Greetings,

I came up with the following two coprime polynomials

a(n)= 9728*n^2 + 2213*n + 123

b(n)=5267712*n^4 + 2587648*n^3 + 480472*n^2 + 39906*n + 1249

gcd((9728*n^2 + 2213*n + 123),
 (5267712*n^4 + 2587648*n^3 + 480472*n^2+39906*n + 1249))  = 1

 j=[];for (n=0,80,j=concat(j, gcd(a(n),b(n))));j
 = [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1,1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]

It appears based on my computations up to n=30,000
 that there is no such n, when a(n) and b(n) are
simultaneously  both primes.

Could someone find the counter-example at higher n's or else
is this conjecture of mine could be proved analytically ?

Regards,
Alex