Timeline for Are concatenations of two consecutive Mersenne numbers which are congruent to 6 mod 7 necessarily composite?
Current License: CC BY-SA 4.0
19 events
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| S Mar 6, 2020 at 22:27 | history | suggested | J. W. Tanner | CC BY-SA 4.0 |
changed palindromatic to palindromic
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| Mar 6, 2020 at 16:34 | review | Suggested edits | |||
| S Mar 6, 2020 at 22:27 | |||||
| Oct 26, 2018 at 20:20 | comment | added | Robert Frost | Incidentally; as to the question of whether concatenation in base 10 could be a special case, one should observe the role of the Golden ratio in Fibonacci (and more generally Lucas) sequences, which is of course based upon the numbers $2$ and $\sqrt 5$, introducing the (albeit highly speculative) but definite possibility that the primes $2,5$ may share some special relationship when these sequences are concatenated. | |
| Oct 26, 2018 at 20:19 | comment | added | Robert Frost | @EnzoCreti In my opinion you were too quick to be intimidated on the question of Lucas sequences. The record should show that your hunch was right because the Mersenne numbers $M_n: n$ not necessarily prime, are a Lucas sequence with recurrence relation $x_{n+1}=2x_n+1$. In fact Aaron's generalisation to repunits base $x$ is in general exactly the Lucas sequence $U_n(x+1,x)$. Your numbers are then the concatenation in base 10 of two terms $U_n, U_{n+1}$ drawn from the same Lucas sequence. | |
| Sep 7, 2018 at 19:19 | comment | added | Aaron Meyerowitz | @EnzoCreti I again suggest investigating the same question for bases other than $c=10$ and also with numbers $\frac{b^k-1}{b-1}$ for $b$ other than $2.$ if they all seem skewed or several of them do then maybe you have something, especially if there is a pattern (like when $b$ is prime and divides $c$ then....) I will say (again) that I rather doubt anything like that because it seems an arbitrary mash-up. Why not glue them together in the other order? and/or write the one in front backwards? and/or interpret the result as a base $11$ numeral which misses the digital $X$? | |
| S Sep 5, 2018 at 13:28 | history | mod moved comments to chat | |||
| S Sep 5, 2018 at 13:28 | comment | added | Ben Webster♦ | Comments are not for extended discussion; this conversation has been moved to chat. | |
| Sep 5, 2018 at 7:19 | comment | added | Gerry Myerson | @Enzo, good – and thanks for deleting the place where you had earlier tried to bring Lucas into it. | |
| Sep 5, 2018 at 7:07 | comment | added | Gerry Myerson | @Enzo, WHAT DOES IT HAVE TO DO WITH LUCAS? | |
| Sep 4, 2018 at 21:27 | comment | added | Gerry Myerson | 91 is divisible by 7, 10001 is divisible by 73, 1875643000 is divisible by 8 – I can play this game too, but you are not telling me what any of this has to do with Lucas? | |
| Sep 4, 2018 at 10:18 | comment | added | Gerry Myerson | OK. And what does this have to do with Lucas? | |
| Sep 4, 2018 at 9:10 | comment | added | Gerry Myerson | I don't know what you mean by "the sixth prime has exponent divisible by 6." What do you mean by the exponent of a prime? I also don't know what you mean by "Do these numbers satisfy a Lucas sequence?" I think I know what a Lucas sequence is, but I don't know which numbers you're talking about, nor what makes you think they might have anything to do with a Lucas sequence. | |
| Aug 27, 2018 at 9:31 | comment | added | Enzo Creti | distribution of residues (mod 7) up to k=366800: [3,6,17,3,8,0] | |
| Jul 17, 2018 at 13:16 | comment | added | Aaron Meyerowitz | @EnzoCreti Maybe try other mixes of bases. If I have it right then the probability that none of these first $35$ would be $6 \bmod 7$ is less than getting $6$ heads in a row from a fair coin but more than that of getting $7$ in a row. | |
| Mar 9, 2018 at 23:24 | history | edited | Aaron Meyerowitz | CC BY-SA 3.0 |
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| Mar 9, 2018 at 22:58 | history | edited | Aaron Meyerowitz | CC BY-SA 3.0 |
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| Mar 7, 2018 at 10:56 | comment | added | Peter | Thanks for your answer. r.e.s (who checked the range upto 10^5) found some further primes, unfortunately, the list has been deleted. It is true that the number of primes found is not significant, and I also think that it is pure coincidence. I just asked this question to see whether I have overlooked something. | |
| Mar 6, 2018 at 21:21 | history | edited | Aaron Meyerowitz | CC BY-SA 3.0 |
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| Mar 6, 2018 at 21:13 | history | answered | Aaron Meyerowitz | CC BY-SA 3.0 |