Timeline for Infinitely many primes of the form $2^n+c$ as $n$ varies?
Current License: CC BY-SA 3.0
18 events
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| Jun 5, 2022 at 7:45 | comment | added | Watson | According to Ofir Gorodetsky, « Fermat primes are special because they have the shape $2^n\pm 1$. More generally, $2^n\pm a^n$ enjoy special structure related to the decomposition of the polynomial $x^n\pm a^n$. For $2^n+3$ there is no such structure to exploit, and in absence of structure, we believe that the behavior is of a random number, in the sense that $2^n+3$ is prime with probability $\sim 1/\log(2^n) \sim 1/n$, as the PNT predicts, and the divergence of $\sum_n 1/n$ suggests that there are infinitely many primes $2^n+3$ (using the 2nd Borel–Cantelli lemma). » | |
| Jun 5, 2022 at 7:43 | comment | added | Watson | (Note that it is nowadays NOT believed that for every Sierpinski number k, there is a finite set of primes P = P_k such that 2^n + k is divisible by some element of P). Also, the heuristic for infinitude of Fermat primes has been removed from Wikpedia in 2019. | |
| Jun 5, 2022 at 7:43 | comment | added | Watson | In the book "Number Theory Revealed, A Masterclass" by Granville, it is written: Calculations suggest there might be infinitely many primes of the form 2^n − 3, though we cannot compute very far since the numbers grow so rapidly. The most optimistic conjecture would be: Fix integer k. • Either there is a finite set of primes P such that for every positive integer n, the number 2^n + k is divisible by some element of P (and so is not prime). • Or there are infinitely many positive integers n for which 2n + k is prime. | |
| Dec 30, 2016 at 8:38 | history | edited | Ben McKay | CC BY-SA 3.0 |
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| Oct 22, 2011 at 8:37 | comment | added | Kevin Buzzard | Yes but it's an easy exercise to check that if $2^n+1$ is prime then it's a Fermat prime. | |
| Oct 21, 2011 at 1:21 | comment | added | Kevin O'Bryant | Fermat primes have the form $2^{2^n}+1$. | |
| Oct 20, 2011 at 7:20 | answer | added | joro | timeline score: 0 | |
| Oct 20, 2011 at 5:22 | answer | added | Timothy Foo | timeline score: 3 | |
| Nov 13, 2009 at 16:04 | history | edited | Kevin Buzzard | CC BY-SA 2.5 |
typo
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| Nov 13, 2009 at 15:38 | vote | accept | Kevin Buzzard | ||
| Nov 13, 2009 at 15:31 | history | edited | Anton Geraschenko |
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| Nov 13, 2009 at 15:11 | answer | added | user631 | timeline score: 34 | |
| Nov 13, 2009 at 14:03 | answer | added | Jaime Montuerto | timeline score: -1 | |
| Nov 13, 2009 at 12:15 | answer | added | Harrison Brown | timeline score: 4 | |
| Nov 13, 2009 at 11:59 | history | edited | Kevin Buzzard | CC BY-SA 2.5 |
Added comment that it was worth reading what Anton says at 5191.
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| Nov 13, 2009 at 10:09 | history | edited | Kevin Buzzard | CC BY-SA 2.5 |
added comment about plausibility of infinitely many Fermat primes
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| Nov 13, 2009 at 9:08 | answer | added | Boris Bukh | timeline score: 1 | |
| Nov 13, 2009 at 7:46 | history | asked | Kevin Buzzard | CC BY-SA 2.5 |