1

Mr. David Norata of south Borneo conjectures that $16^{317}+1$ is the last/largest semi-prime of the form $16^n+1$. But he didn't give any clue why he is so sure about this, he just said that he has a rough proof.

Note that $16^n+1$ can be a semi-prime if and only if $n$ is a prime number, and some values of $n$ for which $16^n+1$ is a semi-prime are $3,5,7,23,37,89,149,173,251,307, 317, \ldots$.

Can you prove or disprove Mr. Norata's conjecture?

Caleb Stanford
  • 45,895
mamihlapinatapai
  • 11
  • Use LATEX please. – Rajat Sep 07 '15 at 13:11
  • 2
    It's unclear how you might expect a concise treatment of such a question. You give no actionable reference by which an interested Reader might assess for themselves a work by "Mr David Norata of [S]outh Borneo", by which you presumably mean the Indonesian portion of the island. – hardmath Sep 07 '15 at 13:12
  • @Rajada, this is the first time for me to post question to this site – mamihlapinatapai Sep 07 '15 at 13:13
  • Okay I will try to edit it. – Rajat Sep 07 '15 at 13:14
  • 2
    One may note that $16^n+1$ is divisible by $17$ for all odd $n$. I find the conjecture dubious, however, being given no reason to believe it. – Milo Brandt Sep 07 '15 at 13:23
  • I'm also really unclear as what you would expect this question to amount to. Do you want someone to prove or disprove someone's suggested conjecture? This is not really what this website is for. – Asaf Karagila Sep 07 '15 at 13:23
  • 1
    Googling David Norata pretty much just leads back to this post. Could the original asker by any chance be the mysterious Mr. Norata? – Forgottenscience Sep 07 '15 at 14:20
  • 1
    The claim that $16^n + 1$ can be semiprime only if $n$ is prime is incorrect unless $n$ is assumed odd. The fifth Fermat number $2^{2^5} + 1 = 16^8 + 1$ has the two prime factors $641$ and $6700417$ (Euler), and other examples are found in the Wikipedia article on Fermat numbers. – hardmath Sep 07 '15 at 14:31
  • 1
    Why does anyone care who Mr Norata is? It's just a math question. – Fixee Sep 07 '15 at 14:59
  • The primality of $~\dfrac{16^{317}+1}{17}~$ can be verified using the elliptic curve method. – Lucian Sep 07 '15 at 20:26

2 Answers2

8

Mr. David Norata's conjecture is false, since $16^{956}+1 = 65537\cdot\text{P1147}.$

Charles
  • 32,122
0

I can confirm via numerical testing that $(16^n+1)/17$ is prime for odd $n<2000$ only if $n=3,5,7,23,37,89,149,173,251,307,317$.

Fixee
  • 11,565