Timeline for Primes arising from permutations
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 15, 2018 at 15:59 | comment | added | Zhi-Wei Sun | @François Brunault Many thanks for your interests and verification! | |
| Nov 15, 2018 at 13:50 | comment | added | François Brunault | I checked and your Conjecture (i) holds for all $n \leq 1000$ with the possible exceptions $n=511, 512, 513, 514, 515, 686, 864, 866, 913, 914, 915, 916, 917, 971, 972, 973, 974, 975, 976$ (it is almost certainly true also for these values, but the permanents get somewhat big). | |
| Nov 15, 2018 at 10:16 | comment | added | François Brunault | More terms: $a(n)=1,2,1,6,1,24,9,38,36,702,196,7386,58^2,69582,213^2,885360,332^2,14335236,800^2,19867008,3318^2,1288115340,13729^2,17909627257,67477^2,363106696516,386492^2,11141446425852,995431^2,371060259505399,4064048^2,1479426535706319,22298599^2,102319410607145600,180526252^2,12597253470226980096,1020038818^2,95009300538155032916...$ (computed up to $n=47$). | |
| Nov 15, 2018 at 9:44 | comment | added | François Brunault | Letting $a(2n+1)=b(n)^2$, the following Pari/GP code works well: b(n)=matpermanent(matrix(n,n,i,j,isprime(2*i*(2*j+1)+1))); it computes $b(n)$ for all $n<30$ in less than 8 minutes on my machine. | |
| Nov 14, 2018 at 23:22 | comment | added | Sylvain JULIEN | Maybe squares of terms of oeis.org/… | |
| Nov 14, 2018 at 23:17 | comment | added | Sylvain JULIEN | Maybe you can also try to prove that $ a(2l+1) $ for non negative integer $ l $ is a square. | |
| Nov 14, 2018 at 23:13 | comment | added | Sylvain JULIEN | Your second conjecture boils down to saying there is a prime $ q_{n,k} $ such that $\tau(k) $ is the inverse of $ k $ in $ \mathbb{Z}/q_{n,k}\mathbb{Z} $ . | |
| Nov 14, 2018 at 16:47 | comment | added | Zhi-Wei Sun | Let $a(n)$ be the number of permutations $\sigma_n$ of $\{1,\ldots,n\}$ such that $k\sigma_n(k)+1$ is prime for every $k=1,\ldots,n$. Then the values of $a(1),\ldots,a(11)$ are $1, \,2,\, 1,\, 6,\, 1,\, 24,\, 9,\, 38,\, 36, \,702, \,196$ respectively. See oeis.org/A321597. | |
| Nov 14, 2018 at 11:16 | answer | added | Christian Elsholtz | timeline score: 12 | |
| Nov 14, 2018 at 0:29 | history | asked | Zhi-Wei Sun | CC BY-SA 4.0 |