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primes of the form (x+1)^p-x^p   Topic List   < Prev Topic  |  Next Topic >
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Re: primes of the form (x+1)^p-x^p

> Format: (n, number of m less than n such that 2^m+3^n or 2^n+3^m is prime)
>
> (1,1) (2,2) (3,2) (4,3) (5,3) (6,3) (9,3) (11,3) (25,2)
> (33,3) (34,3) (54,1) (69,2) (70,3) (97,3) (103,3) (115,3)
> (...)
> (951,3) (958,3) (966,1) (970,3)

Very cool. To notch it up on the coolness factor one could submit this sequence
to OEIS.

Modulo correcting "less than" to "not exceeding" (or specifying that m may be
zero, or correcting a(1)=0 and others in the above).

Actually there are at least 6 sequences:
MU = { n | MU1(n)<=3 } (or why not <= 2 or even <= 1 ?)
MU1(n) = # { m <= n | 2^n+3^m or 2^m+3^n is prime } (or "<" ?)
MU2(n) = min { m | 2^n+3^m or 2^m+3^n is prime }
MU3(n) = max { m <= n | 2^n+3^m or 2^m+3^n is prime } (or "<" ?)
MU4(n) = min { m | 2^n+3^m is prime }
MU5(n) = min { m | 2^m+3^n is prime }


I checked that they are not in OEIS except for MU5 :

A123359 Least m such that 3^n+2^m is prime.

But maybe better double-check...

Maximilian





Wed Apr 8, 2009 2:08 am

maximilian_h...
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Dear prime number fans, is there anything available about possible finiteness of primes of the form (x+1)^p-x^p ? Specifically, some curios reasons led me to...
Maximilian Hasler
maximilian_h...
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Apr 6, 2009
5:16 am

... They are cyclotomic, so have the same kind of rules surrounding admissible factors as Mersennes. (Which are actually quite non-trivial, and the most...
Phil Carmody
thefatphil
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Apr 6, 2009
6:09 am

... Hi Maximilian, p=1399 and 2027 are not the current records for base 7. My personal records are p=69371 and p=86689 for 7^p-6^p. And the largest PRP I have...
j_chrtn
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Apr 6, 2009
5:40 pm

... I have done quite a lot of work on this form, initially summarised in my May 2001 post to the NMBRTHRY list: ...
Mike Oakes
mikeoakes2
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Apr 6, 2009
6:29 pm

Thanks for all replies and useful information. Indeed I would have looked up Lifchitz' database, but I cannot access it these days (or months already...). (It...
Maximilian Hasler
maximilian_h...
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Apr 6, 2009
9:44 pm

... Very cool. To notch it up on the coolness factor one could express the prime exponent 336353 as 3*(2^(2^3+3^2) - 3^(3^2) + 3^(2*3)) - 1 ... On a not too...
Mark Underwood
marku606
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Apr 7, 2009
11:18 pm

... Very cool. To notch it up on the coolness factor one could submit this sequence to OEIS. Modulo correcting "less than" to "not exceeding" (or specifying...
Maximilian Hasler
maximilian_h...
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Apr 8, 2009
2:09 am

... Thank you Maximilian, The OESIS thing would only be appealing to me if the minimum m that caused 2^n+3^m or 2^m+3^n to be prime was always found to be less...
Mark Underwood
marku606
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Apr 9, 2009
9:51 am

... Your conjecture first fails for n = 1679. Best regards David...
David Broadhurst
djbroadhurst
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Apr 9, 2009
11:37 am

Mark Underwood's most interesting sequence for the OEIS might be the cases for which his conjecture fails, namely "Numbers n such that there are no primes of...
David Broadhurst
djbroadhurst
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Apr 9, 2009
1:15 pm

... Thank you David. At least I go into Good Friday with one less potential illusion. :) And that 1679,1743, .. sequence, it is not only hard computing, but...
Mark Underwood
marku606
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Apr 9, 2009
2:05 pm

... "Des goûts et des couleurs, on ne discute pas." [About tastes and colours, one does not argue.] However, I remark that my sequence of blanks is *easier* ...
David Broadhurst
djbroadhurst
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Apr 9, 2009
4:01 pm

... Other such numbers are n = 5314 and n = 6100, but PFGW is still running tests to determine whether there might be any more with n < 6100, so the sequence...
David Broadhurst
djbroadhurst
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Apr 11, 2009
1:41 am

... These numbers include 1679, 1743, 4980, 4982, 5314, 5513, 5695, 6100, 7251, 8218 but my coverage of the range n < 8218 is not complete, so this is not yet...
David Broadhurst
djbroadhurst
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Apr 11, 2009
2:17 pm

... Wow, I am surprised you could go so high, so quickly. Very nice. Some days after you presented the first two numbers, 1679 and 1743, it occurred to me that...
Mark Underwood
marku606
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Apr 13, 2009
5:40 pm

... So far I have 1679, 1743, 4980, 4982, 5314, 5513, 5695, 6100, 6578, 7251, 7406, 7642, 8218, 8331, 9475, 9578 but this is still not a sequence, as there are...
David Broadhurst
djbroadhurst
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Apr 13, 2009
7:00 pm

... 1) Put this script in a file called "loop.txt": SCRIPT DIM bot,1678 DIM top,1680 DIM aa DIM bb DIM nn DIMS st SET aa,bot-1 LABEL loopa SET aa,aa+1 IF aa >...
David Broadhurst
djbroadhurst
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Apr 18, 2009
12:37 pm

Submitted to OEIS: Numbers n such that 2^x + 3^y is never prime when max(x,y) = n 1679, 1743, 4980, 4982, 5314, 5513, 5695, 6100, 6578, 7251, 7406, 7642, 8218,...
David Broadhurst
djbroadhurst
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Apr 17, 2009
2:58 pm

... This form invites possibly the most bizarre, and remarkably efficient, sieve algorithm I've yet had the misfortune of considering. Good job I'm not coding...
Phil Carmody
thefatphil
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Apr 18, 2009
9:33 am

... I have today uploaded a file containing the results of my search for all primes of this form for 2<=b<=1000, 2<=p<10000, done in the years 2000-2008. It is...
Mike Oakes
mikeoakes2
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May 7, 2009
3:29 pm

... Here is a simple link to Mike's interesting table: http://tinyurl.com/d3nf9w David...
David Broadhurst
djbroadhurst
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May 7, 2009
7:10 pm

... (Thanks, Mike !) ... Thanks, David! (Remark: These tiny urls are nice, but can be quite annoying when they point to a website that re-arranged its...
Maximilian Hasler
maximilian_h...
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May 7, 2009
7:51 pm
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