[seqfan] Re: Conjecture re the specific pair of coprime polynomials

[Max Alekseyev]

It is easy to see that a(n)*b(n) is divisible by 3 for any n.
Therefore, both a(n) and b(n) cannot be simultaneously equal to prime numbers.

Regards,
Max

On Fri, Nov 21, 2008 at 2:58 PM, Alexander Povolotsky
<apovolot at gmail.com> wrote:
> 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
>
>
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