[math-fun] Purported proof that there are no odd perfect numbers
In OEIS, the entry A091490 lists prime numbers p such that p^2 + p + 1 has no prime factor larger than p. The entry has a link to http://arxiv.org/abs/hep-th/0401052, a paper by Simon Davis with the intriguing title, *A Proof of the Odd Perfect Number Conjecture*. I looked at the paper for maybe five minutes, and note two things. He really does claim to have proven the conjecture; the only slightly crankish thing I could see was that he calls numbers of the form 1 + p + p^2 + ... + p^n "repunits"; this invocation of base representation is often a calling-card of the amateur. There is some incoherent discussion of the paper on the talk page of the Wikipedia article "Perfect number", with one editor saying that the argument "falls apart" on the second page. Has anyone in this august company looked at Davis's attempt, and/or communicated with Davis on the subject?
The entry has a link to http://arxiv.org/abs/hep-th/0401052, a paper by Simon Davis...
A Robin Chapman had a few comments about this paper when it was published, calling it "very ropey": https://groups.google.com/forum/#!topic/sci.math/CPnCgxBE6Ek
That Robin Chapman is apparently a very good mathematician. Plus for a number of years, and for all I know still, he solved almost every problem in the Monthly each month. —Dan
On Jul 22, 2015, at 8:56 AM, Hans Havermann <gladhobo@teksavvy.com> wrote:
The entry has a link to http://arxiv.org/abs/hep-th/0401052, a paper by Simon Davis...
A Robin Chapman had a few comments about this paper when it was published, calling it "very ropey":
https://groups.google.com/forum/#!topic/sci.math/CPnCgxBE6Ek
http://empslocal.ex.ac.uk/people/staff/rjchapma/rjc.html On Wed, Jul 22, 2015 at 9:01 AM, Dan Asimov <asimov@msri.org> wrote:
That Robin Chapman is apparently a very good mathematician. Plus for a number of years, and for all I know still, he solved almost every problem in the Monthly each month.
—Dan
On Jul 22, 2015, at 8:56 AM, Hans Havermann <gladhobo@teksavvy.com> wrote:
The entry has a link to http://arxiv.org/abs/hep-th/0401052, a paper by Simon Davis...
A Robin Chapman had a few comments about this paper when it was published, calling it "very ropey":
https://groups.google.com/forum/#!topic/sci.math/CPnCgxBE6Ek
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-- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
Seems like a stellar chap. On Wed, Jul 22, 2015 at 12:45 PM, Mike Stay <metaweta@gmail.com> wrote:
http://empslocal.ex.ac.uk/people/staff/rjchapma/rjc.html
On Wed, Jul 22, 2015 at 9:01 AM, Dan Asimov <asimov@msri.org> wrote:
That Robin Chapman is apparently a very good mathematician. Plus for a number of years, and for all I know still, he solved almost every problem in the Monthly each month.
—Dan
On Jul 22, 2015, at 8:56 AM, Hans Havermann <gladhobo@teksavvy.com> wrote:
The entry has a link to http://arxiv.org/abs/hep-th/0401052, a paper by Simon Davis...
A Robin Chapman had a few comments about this paper when it was published, calling it "very ropey":
https://groups.google.com/forum/#!topic/sci.math/CPnCgxBE6Ek
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-- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
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http://empslocal.ex.ac.uk/people/staff/rjchapma/rjc.html On Wed, Jul 22, 2015 at 12:03 PM, Allan Wechsler <acwacw@gmail.com> wrote:
Seems like a stellar chap.
On Wed, Jul 22, 2015 at 12:45 PM, Mike Stay <metaweta@gmail.com> wrote:
http://empslocal.ex.ac.uk/people/staff/rjchapma/rjc.html
On Wed, Jul 22, 2015 at 9:01 AM, Dan Asimov <asimov@msri.org> wrote:
That Robin Chapman is apparently a very good mathematician. Plus for a number of years, and for all I know still, he solved almost every problem in the Monthly each month.
—Dan
On Jul 22, 2015, at 8:56 AM, Hans Havermann <gladhobo@teksavvy.com> wrote:
The entry has a link to http://arxiv.org/abs/hep-th/0401052, a paper by Simon Davis...
A Robin Chapman had a few comments about this paper when it was published, calling it "very ropey":
https://groups.google.com/forum/#!topic/sci.math/CPnCgxBE6Ek
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-- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
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Consider the sequence s_n := (4^n-1)/3, n = 1,2,3,.... Back of the envelope shows that at least for very low n, s_n is squarefree and always has a prime factor that's not a factor of any previous s_n. Do these patterns continue forever, and if so, why? This is OEIS A002450 <https://oeis.org/A002450>, but these features are not mentioned there — so it seems likely they're both false. —Dan
Similar behaviour is common in GFSR sequences over the integers, eg. Mersenne numbers, Fibonacci numbers. But I don't know of any qualitative results ... WFL On 7/22/15, Dan Asimov <asimov@msri.org> wrote:
Consider the sequence s_n := (4^n-1)/3, n = 1,2,3,....
Back of the envelope shows that at least for very low n, s_n is squarefree and always has a prime factor that's not a factor of any previous s_n.
Do these patterns continue forever, and if so, why?
This is OEIS A002450 <https://oeis.org/A002450>, but these features are not mentioned there — so it seems likely they're both false.
—Dan
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s_9 = 87381 = 3^2 7^1 19^1 73^1. In general, I think, if p | s_n, then p | s_kn for all k, and in particular, p^2 | s_np. As for the novel factor, I think that's very likely true. I suspect, in fact, that all non-novel factors are instances of the lemmas I just gave, and they numerically can never get up into the 4^n range, so novel factors must take up the slack. On Wed, Jul 22, 2015 at 6:22 PM, Dan Asimov <asimov@msri.org> wrote:
Consider the sequence s_n := (4^n-1)/3, n = 1,2,3,....
Back of the envelope shows that at least for very low n, s_n is squarefree and always has a prime factor that's not a factor of any previous s_n.
Do these patterns continue forever, and if so, why?
This is OEIS A002450 <https://oeis.org/A002450>, but these features are not mentioned there — so it seems likely they're both false.
—Dan
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Precisely one of your two conjectures is correct. For every odd positive integer k, we have 4^phi(k) - 1 is divisible by k (by Euler-Fermat). Hence 4^phi(3k) - 1 is divisible by 3k (since 3k is an odd positive integer!), so s_phi(3k) is divisible by k. In particular, s_18 = 22906492245 is divisible by 9 and thus isn't squarefree. As for each s_n having a new prime factor, that is a special case of a more general result called Zsigmondy's theorem: https://en.wikipedia.org/wiki/Zsigmondy%27s_theorem Sincerely, Adam P. Goucher
Sent: Wednesday, July 22, 2015 at 11:22 PM From: "Dan Asimov" <asimov@msri.org> To: math-fun <math-fun@mailman.xmission.com> Subject: [math-fun] Number theory pattern?
Consider the sequence s_n := (4^n-1)/3, n = 1,2,3,....
Back of the envelope shows that at least for very low n, s_n is squarefree and always has a prime factor that's not a factor of any previous s_n.
Do these patterns continue forever, and if so, why?
This is OEIS A002450 <https://oeis.org/A002450>, but these features are not mentioned there — so it seems likely they're both false.
—Dan
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(4^n-1)/3 is not square free for n = 9, 10, 18, 20, 21, 27, 30, 36, 40, 42, 45, 50, 54, 55, 60, 63, 68, 70, 72, 78, 80, 81, 84, .... On Wed, Jul 22, 2015 at 5:22 PM, Dan Asimov <asimov@msri.org> wrote:
Consider the sequence s_n := (4^n-1)/3, n = 1,2,3,....
Back of the envelope shows that at least for very low n, s_n is squarefree and always has a prime factor that's not a factor of any previous s_n.
Do these patterns continue forever, and if so, why?
This is OEIS A002450 <https://oeis.org/A002450>, but these features are not mentioned there — so it seems likely they're both false.
—Dan
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These are called primitive divisors. There's a theorem of Zsigismondy which covers this . See this paper for details: http://www.uea.ac.uk/~h008/research/primes.pdf Victor Sent from my iPhone
On Jul 22, 2015, at 18:22, Dan Asimov <asimov@msri.org> wrote:
Consider the sequence s_n := (4^n-1)/3, n = 1,2,3,....
Back of the envelope shows that at least for very low n, s_n is squarefree and always has a prime factor that's not a factor of any previous s_n.
Do these patterns continue forever, and if so, why?
This is OEIS A002450 <https://oeis.org/A002450>, but these features are not mentioned there — so it seems likely they're both false.
—Dan
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The primitive divisors of A002450 are listed in A129735 (along with a reference to Zsigi) Best regards Neil Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com On Thu, Jul 23, 2015 at 8:41 AM, Victor S. Miller <victorsmiller@gmail.com> wrote:
These are called primitive divisors. There's a theorem of Zsigismondy which covers this . See this paper for details: http://www.uea.ac.uk/~h008/research/primes.pdf
Victor
Sent from my iPhone
On Jul 22, 2015, at 18:22, Dan Asimov <asimov@msri.org> wrote:
Consider the sequence s_n := (4^n-1)/3, n = 1,2,3,....
Back of the envelope shows that at least for very low n, s_n is squarefree and always has a prime factor that's not a factor of any previous s_n.
Do these patterns continue forever, and if so, why?
This is OEIS A002450 <https://oeis.org/A002450>, but these features are not mentioned there — so it seems likely they're both false.
—Dan
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Thanks for all the fascinating stuff I knew nothing about. Incidentally, the list of primitive prime divisors (primes that don't divide any previous term of s_n = (4^n-1)/3) in A129735 raises a question or two whose answers I may have missed: 1) Does every prime appear in that list? (A129735 seems to include every prime through at least 53.) 2) Can a new term ever include more than one primitive prime factor? —Dan
On Jul 23, 2015, at 8:01 AM, Neil Sloane <njasloane@gmail.com> wrote:
The primitive divisors of A002450 are listed in A129735 (along with a reference to Zsigi)
Best regards Neil
Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com
On Thu, Jul 23, 2015 at 8:41 AM, Victor S. Miller <victorsmiller@gmail.com> wrote:
These are called primitive divisors. There's a theorem of Zsigismondy which covers this . See this paper for details: http://www.uea.ac.uk/~h008/research/primes.pdf
Victor
Sent from my iPhone
On Jul 22, 2015, at 18:22, Dan Asimov <asimov@msri.org> wrote:
Consider the sequence s_n := (4^n-1)/3, n = 1,2,3,....
Back of the envelope shows that at least for very low n, s_n is squarefree and always has a prime factor that's not a factor of any previous s_n.
Do these patterns continue forever, and if so, why?
This is OEIS A002450 <https://oeis.org/A002450>, but these features are not mentioned there — so it seems likely they're both false.
—Dan
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1) Yes, that list is by induction the list of primes that divide any s_n, and I've shown that every (odd) integer divides infinitely many s_n. Hence every odd prime occurs as a primitive prime divisor. 2) I see no reason why that shouldn't happen. (Indeed, if that weren't the case, then by (1) we would have some bizarre enumeration of the primes.)
Sent: Thursday, July 23, 2015 at 6:29 PM From: "Dan Asimov" <dasimov@earthlink.net> To: math-fun <math-fun@mailman.xmission.com> Subject: Re: [math-fun] Number theory pattern?
Thanks for all the fascinating stuff I knew nothing about.
Incidentally, the list of primitive prime divisors (primes that don't divide any previous term of s_n = (4^n-1)/3) in A129735 raises a question or two whose answers I may have missed:
1) Does every prime appear in that list? (A129735 seems to include every prime through at least 53.)
2) Can a new term ever include more than one primitive prime factor?
—Dan
On Jul 23, 2015, at 8:01 AM, Neil Sloane <njasloane@gmail.com> wrote:
The primitive divisors of A002450 are listed in A129735 (along with a reference to Zsigi)
Best regards Neil
Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com
On Thu, Jul 23, 2015 at 8:41 AM, Victor S. Miller <victorsmiller@gmail.com> wrote:
These are called primitive divisors. There's a theorem of Zsigismondy which covers this . See this paper for details: http://www.uea.ac.uk/~h008/research/primes.pdf
Victor
Sent from my iPhone
On Jul 22, 2015, at 18:22, Dan Asimov <asimov@msri.org> wrote:
Consider the sequence s_n := (4^n-1)/3, n = 1,2,3,....
Back of the envelope shows that at least for very low n, s_n is squarefree and always has a prime factor that's not a factor of any previous s_n.
Do these patterns continue forever, and if so, why?
This is OEIS A002450 <https://oeis.org/A002450>, but these features are not mentioned there — so it seems likely they're both false.
—Dan
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s_3 = 21 introduces 3 and 7 -- isn't that an example of Dan's (2)? s_5 = 341 = 11*31; isn't that another example of two primes making a simultaneous debut? On Thu, Jul 23, 2015 at 2:10 PM, Adam P. Goucher <apgoucher@gmx.com> wrote:
1) Yes, that list is by induction the list of primes that divide any s_n, and I've shown that every (odd) integer divides infinitely many s_n. Hence every odd prime occurs as a primitive prime divisor.
2) I see no reason why that shouldn't happen. (Indeed, if that weren't the case, then by (1) we would have some bizarre enumeration of the primes.)
Sent: Thursday, July 23, 2015 at 6:29 PM From: "Dan Asimov" <dasimov@earthlink.net> To: math-fun <math-fun@mailman.xmission.com> Subject: Re: [math-fun] Number theory pattern?
Thanks for all the fascinating stuff I knew nothing about.
Incidentally, the list of primitive prime divisors (primes that don't divide any previous term of s_n = (4^n-1)/3) in A129735 raises a question or two whose answers I may have missed:
1) Does every prime appear in that list? (A129735 seems to include every prime through at least 53.)
2) Can a new term ever include more than one primitive prime factor?
—Dan
On Jul 23, 2015, at 8:01 AM, Neil Sloane <njasloane@gmail.com> wrote:
The primitive divisors of A002450 are listed in A129735 (along with a reference to Zsigi)
Best regards Neil
Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com
On Thu, Jul 23, 2015 at 8:41 AM, Victor S. Miller < victorsmiller@gmail.com> wrote:
These are called primitive divisors. There's a theorem of Zsigismondy which covers this . See this paper for details: http://www.uea.ac.uk/~h008/research/primes.pdf
Victor
Sent from my iPhone
On Jul 22, 2015, at 18:22, Dan Asimov <asimov@msri.org> wrote:
Consider the sequence s_n := (4^n-1)/3, n = 1,2,3,....
Back of the envelope shows that at least for very low n, s_n is squarefree and always has a prime factor that's not a factor of any previous s_n.
Do these patterns continue forever, and if so, why?
This is OEIS A002450 <https://oeis.org/A002450>, but these features are not mentioned there — so it seems likely they're both false.
—Dan
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If the new primes appeared one at a time, it would make the task of Factoring Mersenne numbers much easier: To factor Mn, just test for & Remove all the earlier divisors of Msmaller; any leftover piece would be Prime. No such luck. To build on the 2^11-1 = 23.89 example ... 4^11-1 = 2049.2047 = 3.683.23.89, so the sequence (4^N-1)/3 will have First appearances of 23, 89, and 683 at term N=11. A similar pattern arises for 4^Prime-1, where all the new prime divisors Are of the shape 2 K Prime + 1. (23 & 89 are examples with Prime=11.) Rich -----Original Message----- From: math-fun [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of Allan Wechsler Sent: Thursday, July 23, 2015 12:38 PM To: math-fun Subject: [EXTERNAL] Re: [math-fun] Number theory pattern? s_3 = 21 introduces 3 and 7 -- isn't that an example of Dan's (2)? s_5 = 341 = 11*31; isn't that another example of two primes making a simultaneous debut? On Thu, Jul 23, 2015 at 2:10 PM, Adam P. Goucher <apgoucher@gmx.com> wrote:
1) Yes, that list is by induction the list of primes that divide any s_n, and I've shown that every (odd) integer divides infinitely many s_n. Hence every odd prime occurs as a primitive prime divisor.
2) I see no reason why that shouldn't happen. (Indeed, if that weren't the case, then by (1) we would have some bizarre enumeration of the primes.)
Sent: Thursday, July 23, 2015 at 6:29 PM From: "Dan Asimov" <dasimov@earthlink.net> To: math-fun <math-fun@mailman.xmission.com> Subject: Re: [math-fun] Number theory pattern?
Thanks for all the fascinating stuff I knew nothing about.
Incidentally, the list of primitive prime divisors (primes that don't divide any previous term of s_n = (4^n-1)/3) in A129735 raises a question or two whose answers I may have missed:
1) Does every prime appear in that list? (A129735 seems to include every prime through at least 53.)
2) Can a new term ever include more than one primitive prime factor?
—Dan
On Jul 23, 2015, at 8:01 AM, Neil Sloane <njasloane@gmail.com> wrote:
The primitive divisors of A002450 are listed in A129735 (along with a reference to Zsigi)
Best regards Neil
Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com
On Thu, Jul 23, 2015 at 8:41 AM, Victor S. Miller < victorsmiller@gmail.com> wrote:
These are called primitive divisors. There's a theorem of Zsigismondy which covers this . See this paper for details: http://www.uea.ac.uk/~h008/research/primes.pdf
Victor
Sent from my iPhone
On Jul 22, 2015, at 18:22, Dan Asimov <asimov@msri.org> wrote:
Consider the sequence s_n := (4^n-1)/3, n = 1,2,3,....
Back of the envelope shows that at least for very low n, s_n is squarefree and always has a prime factor that's not a factor of any previous s_n.
Do these patterns continue forever, and if so, why?
This is OEIS A002450 <https://oeis.org/A002450>, but these features are not mentioned there — so it seems likely they're both false.
—Dan
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Yes, it is. After that they seem to enter one by one. —Dan
On Jul 23, 2015, at 11:38 AM, Allan Wechsler <acwacw@gmail.com> wrote:
s_3 = 21 introduces 3 and 7 -- isn't that an example of Dan's (2)? s_5 = 341 = 11*31; isn't that another example of two primes making a simultaneous debut?
On Thu, Jul 23, 2015 at 2:10 PM, Adam P. Goucher <apgoucher@gmx.com> wrote:
1) Yes, that list is by induction the list of primes that divide any s_n, and I've shown that every (odd) integer divides infinitely many s_n. Hence every odd prime occurs as a primitive prime divisor.
2) I see no reason why that shouldn't happen. (Indeed, if that weren't the case, then by (1) we would have some bizarre enumeration of the primes.)
Sent: Thursday, July 23, 2015 at 6:29 PM From: "Dan Asimov" <dasimov@earthlink.net> To: math-fun <math-fun@mailman.xmission.com> Subject: Re: [math-fun] Number theory pattern?
Thanks for all the fascinating stuff I knew nothing about.
Incidentally, the list of primitive prime divisors (primes that don't divide any previous term of s_n = (4^n-1)/3) in A129735 raises a question or two whose answers I may have missed:
1) Does every prime appear in that list? (A129735 seems to include every prime through at least 53.)
2) Can a new term ever include more than one primitive prime factor?
—Dan
On Jul 23, 2015, at 8:01 AM, Neil Sloane <njasloane@gmail.com> wrote:
The primitive divisors of A002450 are listed in A129735 (along with a reference to Zsigi)
Best regards Neil
Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com
On Thu, Jul 23, 2015 at 8:41 AM, Victor S. Miller < victorsmiller@gmail.com> wrote:
These are called primitive divisors. There's a theorem of Zsigismondy which covers this . See this paper for details: http://www.uea.ac.uk/~h008/research/primes.pdf
Victor
Sent from my iPhone
On Jul 22, 2015, at 18:22, Dan Asimov <asimov@msri.org> wrote:
Consider the sequence s_n := (4^n-1)/3, n = 1,2,3,....
Back of the envelope shows that at least for very low n, s_n is squarefree and always has a prime factor that's not a factor of any previous s_n.
Do these patterns continue forever, and if so, why?
This is OEIS A002450 <https://oeis.org/A002450>, but these features are not mentioned there — so it seems likely they're both false.
—Dan
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As the list goes on there is a tendency for more to enter each time, but not always. It starts as follows: 1, [5] 2, [3, 7] 3, [17] 4, [11, 31] 5, [13] 6, [43, 127] 7, [257] 8, [19, 73] 9, [41] 10, [23, 89, 683] 11, [241] 12, [2731, 8191] 13, [29, 113] 14, [151, 331] 15, [65537] 16, [43691, 131071] 17, [37, 109] 18, [174763, 524287] 19, [61681] 20, [337, 5419] 21, [397, 2113] 22, [47, 178481, 2796203] 23, [97, 673] 24, [251, 601, 1801, 4051] On Thu, Jul 23, 2015 at 2:14 PM, Dan Asimov <asimov@msri.org> wrote:
Yes, it is. After that they seem to enter one by one.
—Dan
On Jul 23, 2015, at 11:38 AM, Allan Wechsler <acwacw@gmail.com> wrote:
s_3 = 21 introduces 3 and 7 -- isn't that an example of Dan's (2)? s_5 = 341 = 11*31; isn't that another example of two primes making a simultaneous debut?
On Thu, Jul 23, 2015 at 2:10 PM, Adam P. Goucher <apgoucher@gmx.com> wrote:
1) Yes, that list is by induction the list of primes that divide any s_n, and I've shown that every (odd) integer divides infinitely many s_n. Hence every odd prime occurs as a primitive prime divisor.
2) I see no reason why that shouldn't happen. (Indeed, if that weren't the case, then by (1) we would have some bizarre enumeration of the primes.)
Sent: Thursday, July 23, 2015 at 6:29 PM From: "Dan Asimov" <dasimov@earthlink.net> To: math-fun <math-fun@mailman.xmission.com> Subject: Re: [math-fun] Number theory pattern?
Thanks for all the fascinating stuff I knew nothing about.
Incidentally, the list of primitive prime divisors (primes that don't divide any previous term of s_n = (4^n-1)/3) in A129735 raises a question or two whose answers I may have missed:
1) Does every prime appear in that list? (A129735 seems to include every prime through at least 53.)
2) Can a new term ever include more than one primitive prime factor?
—Dan
On Jul 23, 2015, at 8:01 AM, Neil Sloane <njasloane@gmail.com> wrote:
The primitive divisors of A002450 are listed in A129735 (along with a reference to Zsigi)
Best regards Neil
Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com
On Thu, Jul 23, 2015 at 8:41 AM, Victor S. Miller < victorsmiller@gmail.com> wrote:
These are called primitive divisors. There's a theorem of Zsigismondy which covers this . See this paper for details: http://www.uea.ac.uk/~h008/research/primes.pdf
Victor
Sent from my iPhone
> On Jul 22, 2015, at 18:22, Dan Asimov <asimov@msri.org> wrote: > > Consider the sequence s_n := (4^n-1)/3, n = 1,2,3,.... > > Back of the envelope shows that at least for very low n, s_n is squarefree and always has a prime factor that's not a factor of any previous s_n. > > Do these patterns continue forever, and if so, why? > > This is OEIS A002450 <https://oeis.org/A002450>, but these features are not mentioned there — so it seems likely they're both false. > > —Dan > > > _______________________________________________ > math-fun mailing list > math-fun@mailman.xmission.com > https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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(1) Yes. Suppose you want to know the first element A002450 that is divisible by 59. Calculate the powers of 4 mod 59. They go: 1, 4, 16, 5, 10, ... and eventually they must cycle back to 1 because of general Fermat stuff. Certainly 4^58 = 1 mod 59. So 4^58-1 is a multiple of 59, and so is (4^58-1)/3. (2) My intuition is yes. Certainly in other similar sequences (the Mersenne numbers, for example) you can get multiple novel primes. 2^11-1, for instance, is 23*89. I don't know how to construct an example for A002450, though. On Thu, Jul 23, 2015 at 1:29 PM, Dan Asimov <dasimov@earthlink.net> wrote:
Thanks for all the fascinating stuff I knew nothing about.
Incidentally, the list of primitive prime divisors (primes that don't divide any previous term of s_n = (4^n-1)/3) in A129735 raises a question or two whose answers I may have missed:
1) Does every prime appear in that list? (A129735 seems to include every prime through at least 53.)
2) Can a new term ever include more than one primitive prime factor?
—Dan
On Jul 23, 2015, at 8:01 AM, Neil Sloane <njasloane@gmail.com> wrote:
The primitive divisors of A002450 are listed in A129735 (along with a reference to Zsigi)
Best regards Neil
Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com
On Thu, Jul 23, 2015 at 8:41 AM, Victor S. Miller < victorsmiller@gmail.com> wrote:
These are called primitive divisors. There's a theorem of Zsigismondy which covers this . See this paper for details: http://www.uea.ac.uk/~h008/research/primes.pdf
Victor
Sent from my iPhone
On Jul 22, 2015, at 18:22, Dan Asimov <asimov@msri.org> wrote:
Consider the sequence s_n := (4^n-1)/3, n = 1,2,3,....
Back of the envelope shows that at least for very low n, s_n is squarefree and always has a prime factor that's not a factor of any previous s_n.
Do these patterns continue forever, and if so, why?
This is OEIS A002450 <https://oeis.org/A002450>, but these features are not mentioned there — so it seems likely they're both false.
—Dan
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participants (11)
-
Adam P. Goucher -
Allan Wechsler -
Dan Asimov -
Dan Asimov -
Fred Lunnon -
Hans Havermann -
James Buddenhagen -
Mike Stay -
Neil Sloane -
Schroeppel, Richard -
Victor S. Miller