mersenneforum.org Finite differences for the win
 Register FAQ Search Today's Posts Mark Forums Read

 2018-01-01, 13:00 #1 George M   Dec 2017 1100102 Posts Finite differences for the win I don’t think this goes by the rules because I am not making a guess. I just think a good way to find a probable value for p such that M51 = 2^p - 1 is as follows: Arrange $p_1, p_2,p_3,\ldots p_{50}$ in a sequence just like so where $p_n$ would be a prime p for the n-th mersenne prime. Then, find the difference between two adjacent pairs of those prime numbers $(p_1, p_2), (p_2, p_3), (p_3, p_4)\ldots (p_{49}, p_{50})$ Now apply the same rule to the difference of these prime numbers. Let $a_1 = p_2 - p_1$ and then $a_2 = p_3 - p_2$ and so on. Now, find the difference between $a_1$ and $a_2$ and find the difference of each adjacent pair of $(a_n, a_{n + 1})$ just like what you did with the primes. Keep doing this and eventually you will reach a constant C where the difference between $C_n$ and $C_{n + 1}$ is 0 because $C_n = C_{n + 1}$. Now work your way backwards by assuming that if there existed $p_{51}$ then this process would still reach 0, implying that it reaches C, implying etc... After that, you will reach a value V that is likely to be equal to $p_{51}$. Last fiddled with by George M on 2018-01-01 at 13:01
2018-01-01, 13:41   #2
axn

Jun 2003

2·7·17·23 Posts

Quote:
 Originally Posted by George M I just think a good way to find a probable value for p such that M51 = 2^p - 1 is as follows:
Ok... So you want to do polynomial interpolation?

Code:
v=[2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583, 25964951, 30402457, 32582657, 37156667, 42643801, 43112609, 57885161, 74207281];
polinterpolate(v,,#v+1)
Code:
-4668069701844879790
Now what? Maybe we had too many primes in there. Let's try with > 10m digits
Code:
v=[37156667, 42643801, 43112609, 57885161, 74207281];
polinterpolate(v,,#v+1)
Code:
47248547
Hmmm... Not quite what you'd expect?

Last fiddled with by axn on 2018-01-01 at 13:45 Reason: No -> Now

 2018-01-01, 21:37 #3 Batalov     "Serge" Mar 2008 San Diego, Calif. 101×103 Posts Oh no, this curve is definitely coming down. Deja vu all over again.

 Similar Threads Thread Thread Starter Forum Replies Last Post Trejack Miscellaneous Math 11 2016-05-12 04:15 CRGreathouse Math 3 2009-05-29 20:38 nekketsu Information & Answers 1 2007-12-16 09:13 MooMoo2 Riesel Prime Search 6 2006-09-27 18:51 koekie Software 8 2003-02-01 21:41

All times are UTC. The time now is 14:17.

Sun Oct 1 14:17:51 UTC 2023 up 18 days, 12 hrs, 0 users, load averages: 0.71, 0.77, 0.82