# A routine for testing a conjecture

The ec numbers are so defined:

ec(k) = (2^k-1)*10^d + 2^(k-1) - 1


where d is the number of decimal digits of 2^(k-1) - 1 . In other words these numbers are formed by the base 10 concatenation of two consecutive Mersenne numbers, for example: 157, 12763, 40952047...

For some values of k, ec(k) is probable prime. I found that up to k=565.000 there is no probable prime of the form (2^k-1)*10^d + 2^(k-1) - 1 which is congruent to 6 mod 7. So I conjectured that there is no probable prime of this form congruent to 6 mod 7. Has somebody an efficient program for Sage to test this conjecture further?

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I guess you can just program it to create these numbers (and probably can use a string concatenation to do so more efficiently than multiplication?). But without some extra theory helping reduce the primality testing like we have for Mersenne numbers, it might be hard to make a test that was "efficient" in the sense you probably mean.

( 2018-12-06 17:47:39 +0100 )edit
( 2018-12-07 19:36:31 +0100 )edit

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I tried the following:

sage: R = Zmod(7)
sage: for k in [2..500]:
....:     a = 2^k-1
....:     b = 2^(k-1)-1
....:     N = ZZ('{}{}'.format(a, b))
....:     if R(N) != R(6):
....:         continue
....:     print( "k=%s Is ec(k) prime? %s. Factorization follows:\nec(k) = %s\n"
....:            % (k, N.is_prime(), N.factor()) )
....:
k=10 Is ec(k) prime? False. Factorization follows:
ec(k) = 19 * 103 * 523

k=11 Is ec(k) prime? False. Factorization follows:
ec(k) = 479 * 42737

k=14 Is ec(k) prime? False. Factorization follows:
ec(k) = 11 * 593 * 25117

k=28 Is ec(k) prime? False. Factorization follows:
ec(k) = 233 * 1607 * 716915680417

k=32 Is ec(k) prime? False. Factorization follows:
ec(k) = 131 * 4463 * 21601 * 44623 * 76213

k=49 Is ec(k) prime? False. Factorization follows:
ec(k) = 5 * 757 * 16333 * 1225015921 * 7433549000531

k=53 Is ec(k) prime? False. Factorization follows:
ec(k) = 5 * 337 * 53455188455436151040711945027

k=70 Is ec(k) prime? False. Factorization follows:
ec(k) = 109 * 839 * 75046613 * 241028036131 * 713694876516226387

k=71 Is ec(k) prime? False. Factorization follows:
ec(k) = 23 * 15737 * 65234886529801619745410789282584431073

k=74 Is ec(k) prime? False. Factorization follows:
ec(k) = 11 * 19 * 269 * 9532513 * 352463140718866450408093341421867

k=88 Is ec(k) prime? False. Factorization follows:
ec(k) = 31 * 73875972467027 * 135137137017690741456718218482342349371

k=92 Is ec(k) prime? False. Factorization follows:
ec(k) = 730315371175567 * 39625364799966331 * 1711101949753493724071011

k=109 Is ec(k) prime? False. Factorization follows:
ec(k) = 5 * 653 * 1606053961 * 10568312139584431 * 11711717200756188938696404879503826537

k=113 Is ec(k) prime? False. Factorization follows:
ec(k) = 5 * 193 * 149270993 * 24073195224569 * 29946980304751014175703201587995219695454299

k=130 Is ec(k) prime? False. Factorization follows:
ec(k) = 1129 * 959806273091 * 7986419296370382549203 * 157278660923445868781899626742322195007583

k=131 Is ec(k) prime? False. Factorization follows:
ec(k) = 51162479 * 2784303036149 * 567204394305177089 * 336916099985640327995882896303775632213517

k=148 Is ec(k) prime? False. Factorization follows:
ec(k) = 76292370683 * 46974096157407024149 * 60514961739327090714406687 * 1645269521635269788991753843968263

k=152 Is ec(k) prime? False. Factorization follows:
ec(k) = 31 * 107 * 677 * 2131 * 145361 * 173087 * 452931678706211 * 6576742625936687 * 159179030364736283121060312673245829294207

k=169 Is ec(k) prime? False. Factorization follows:
ec(k) = 5 * 67360183384144337 * 1317991479336685050851 * 1685712081595174413704015925364267420341119063462133679767171233

k=173 Is ec(k) prime? False. Factorization follows:
ec(k) = 5 * 1543 * 11287 * 1374911658072068607645715891596827336333789380835457971697269244791317892283158474011048315274579

k=190 Is ec(k) prime? False. Factorization follows:
ec(k) = 19 * 82593443886666852155734071357995610738188887427158348853883401985101228162919972298836892542211199706871473911269


and i had to stop here. (Since i did not see any sense in finding prime numbers of this "concatenated shape". This is non-structural mathematics for me.)

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@dan_fulea the program checks primes or probable primes? I am looking for probable primes.

( 2018-12-11 16:11:22 +0100 )edit

@dan_fulea and what if I want to cancel the factorization?

( 2018-12-11 18:18:21 +0100 )edit