4

if the prime number such $p\equiv 5\pmod 8,p\neq 29$,I conjecture $$\gcd(p^2+3p+4,2^p-1)=1,p\neq 29$$

I test $p<100$ is true

enter image description here

math110
  • 93,304
  • 5
    The claim is false for $p = 29$ as $\text{gcd}(932,536870911) = 233$ – The Demonix _ Hermit Jan 12 '20 at 05:31
  • 1
    Aside from that , checked for all $p < 10^6$ without any counter example other than $29$ – The Demonix _ Hermit Jan 12 '20 at 05:43
  • 1
    Would you be content with the range, lets say, $p\le 10^9$, or only with a full proof ? – Peter Jan 12 '20 at 10:58
  • I tested up to $1.7\times 10^8$ and didn't find any counterexamples - I'd be willing to believe that there are no counterexamples largely by random chance - if $f$ is any function growing faster than $n^{1+\delta}$ and I pick a random $x_n$ for each $n$, the expected number of times $x_n$ and $f(n)$ share a factor is finite. – Milo Brandt Jan 15 '20 at 01:22

0 Answers0