Timeline for How $a+b$ can grow when $a!b! \mid n!$
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 29, 2013 at 21:53 | comment | added | Noam D. Elkies | Meanwhile $v=16$ turned up: $(13106, 52447, 65536)$. Yes, $n=2^{16}$. What was that about $2$-adic analysis again? | |
| Jul 29, 2013 at 21:40 | comment | added | Will Sawin | @S. Carnahan - can I see the sage code you used? | |
| Jul 29, 2013 at 6:45 | comment | added | Noam D. Elkies | Yes, the $v=15$ example [not $\nu$, since it stands for @Aaron Meyerowitz's "virtue"] just turned up here too, using val(m,p, m1,v) = m1=m; v=0; while(m1>0, m1\=p; v+=m1); return(v) and then test(a,b,n) = for(f=n+1,a+b,F=factor(f)[,1];for(i=1,#F,p=F[i];if(val(n,p<val(a,p)+val(b,p),return(0))));return(1) | |
| Jul 29, 2013 at 6:28 | comment | added | S. Carnahan♦ | for $\nu = 15$, the smallest is $[14335, 18458, 32778]$, and for $\nu = 16$, you get two examples for 32808. (done with SAGE, using $p$-valuations of factorials) | |
| Jul 29, 2013 at 3:30 | comment | added | Will Sawin | For comparison, the $2$-adic best possibles are 1 [1, 1, 1] 2 [3, 3, 4] 3 [4, 7, 8] 4 [5, 7, 8] 5 [6,15, 16] 6 [7, 15, 16] 7 [8, 127, 128] 8 [9, 127, 128] 9 [10, 255, 256] 10 [11, 255, 256] 11 [12, 1023, 1024] 12 [13, 1023, 1024] 13 [14, 2047, 2048] 14 [15, 2047, 2048]. This is evidence against my earlier belief that one can do a lot better than just the 2-adic analysis! | |
| Jul 29, 2013 at 3:16 | comment | added | Noam D. Elkies | (Unless you allow $n=0$, in which case the $v=1$ and $v=2$ records are $(0,1,0)$ and $(1,1,0)$ respectively.) | |
| Jul 29, 2013 at 3:14 | history | answered | Noam D. Elkies | CC BY-SA 3.0 |