[math-fun] Re: your mail
I'll make a wild guess that it can be proved that no such n exists; I'll copy this to some people who may be able to confirm or deny this. R. On Wed, 22 Jan 2003, Mr. Nayandeep Deka Baruah wrote:
Dear Professors Guy and Borwein,
I would like to know from you whether the following result is still a conjecture or has been proved by somebody.
There exists a composite integer n such that for each prime divisor p of n (p+1)|(n+1).
If it is true then what is the smallest such number? Are such numbers are infinitely many?
I would be extremely grateful for your help.
With best regards,
Nayandeep Deka Baruah Dept. of Math. Sciences, Tezpur University Napaam-784028 Assam, INDIA.
n = 8. -----Original Message----- From: math-fun-admin@mailman.xmission.com [mailto:math-fun-admin@mailman.xmission.com]On Behalf Of Richard Guy Sent: Wednesday, January 22, 2003 12:17 To: Mr. Nayandeep Deka Baruah Cc: Jonathan m. Borwein; Math Fun; Number Theory List Subject: [math-fun] Re: your mail I'll make a wild guess that it can be proved that no such n exists; I'll copy this to some people who may be able to confirm or deny this. R. On Wed, 22 Jan 2003, Mr. Nayandeep Deka Baruah wrote:
Dear Professors Guy and Borwein,
I would like to know from you whether the following result is still a conjecture or has been proved by somebody.
There exists a composite integer n such that for each prime divisor p of n (p+1)|(n+1).
If it is true then what is the smallest such number? Are such numbers are infinitely many?
I would be extremely grateful for your help.
With best regards,
Nayandeep Deka Baruah Dept. of Math. Sciences, Tezpur University Napaam-784028 Assam, INDIA.
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Numbers less than 100000 with this property are the following: 8, 27, 32, 63, 125, 128, 243, 275, 343, 399, 512, 567, 575, 935, 1127, 1331, 1539, 2015, 2048, 2187, 2197, 2303, 2783, 2915, 3087, 3125, 4563, 4913, 4991, 5103, 5719, 5831, 6399, 6859, 6875, 6929, 7055, 7139, 7625, 8192, 8855, 12167, 12719, 14027, 14375, 14399, 16807, 17303, 18095, 19683, 20519, 20705, 20999, 21125, 21299, 22847, 24389, 24863, 26999, 27783, 28175, 28223, 29315, 29375, 29791, 31535, 32319, 32375, 32768, 33275, 34775, 36125, 38759, 41327, 45927, 46079, 49247, 49619, 50653, 51359, 55223, 58563, 60059, 60543, 63503, 64619, 67199, 68921, 69575, 73535, 74375, 74431, 75087, 76751, 78125, 79507, 80189, 81719, 86975, 88559, 89375, 89675, 90287 --Ed Pegg Jr --- Richard Guy <rkg@cpsc.ucalgary.ca> wrote:
I'll make a wild guess that it can be proved that no such n exists; I'll copy this to some people who may be able to confirm or deny this.
R.
On Wed, 22 Jan 2003, Mr. Nayandeep Deka Baruah wrote:
Dear Professors Guy and Borwein,
I would like to know from you whether the following result is still a conjecture or has been proved by somebody.
There exists a composite integer n such that for each prime divisor p of n (p+1)|(n+1).
If it is true then what is the smallest such number? Are such numbers are infinitely many?
I would be extremely grateful for your help.
With best regards,
Nayandeep Deka Baruah Dept. of Math. Sciences, Tezpur University Napaam-784028 Assam, INDIA.
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Offhand, I can think of lots of numbers that satisfy the conjecture. Consider a mersenne prime p=2^m-1 for some integer m. Let n=p^(2k+1), and p+1|n+1 since p+1=2^m and n(mod 2^m)=(-1)^(2k+1)+1=0. Gershon Bialer
On Wed, 22 Jan 2003, Mr. Nayandeep Deka Baruah wrote:
Dear Professors Guy and Borwein,
I would like to know from you whether the following result is still a conjecture or has been proved by somebody.
There exists a composite integer n such that for each prime divisor p of n (p+1)|(n+1).
If it is true then what is the smallest such number? Are such numbers are infinitely many?
I would be extremely grateful for your help.
With best regards,
Nayandeep Deka Baruah Dept. of Math. Sciences, Tezpur University Napaam-784028 Assam, INDIA.
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
p^(2k+1) for any prime works, since (x+1) | (x^n + 1) when n is odd. -----Original Message----- From: math-fun-admin@mailman.xmission.com [mailto:math-fun-admin@mailman.xmission.com]On Behalf Of Gershon Bialer Sent: Wednesday, January 22, 2003 12:51 To: math-fun@mailman.xmission.com Subject: Re: [math-fun] Re: your mail Offhand, I can think of lots of numbers that satisfy the conjecture. Consider a mersenne prime p=2^m-1 for some integer m. Let n=p^(2k+1), and p+1|n+1 since p+1=2^m and n(mod 2^m)=(-1)^(2k+1)+1=0. Gershon Bialer
On Wed, 22 Jan 2003, Mr. Nayandeep Deka Baruah wrote:
Dear Professors Guy and Borwein,
I would like to know from you whether the following result is still a conjecture or has been proved by somebody.
There exists a composite integer n such that for each prime divisor p of n (p+1)|(n+1).
If it is true then what is the smallest such number? Are such numbers are infinitely many?
I would be extremely grateful for your help.
With best regards,
Nayandeep Deka Baruah Dept. of Math. Sciences, Tezpur University Napaam-784028 Assam, INDIA.
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Apologies for my wildness. Mike Speciner gives the example n = 8, which generalizes to any odd power of 2. Are these the only examples? Is the sequence 8, 32, 128, 512, 2048, ... (& including any other numbers I've overlooked) in OEIS ? R. On Wed, 22 Jan 2003, Richard Guy wrote:
I'll make a wild guess that it can be proved that no such n exists; I'll copy this to some people who may be able to confirm or deny this.
R.
On Wed, 22 Jan 2003, Mr. Nayandeep Deka Baruah wrote:
Dear Professors Guy and Borwein,
I would like to know from you whether the following result is still a conjecture or has been proved by somebody.
There exists a composite integer n such that for each prime divisor p of n (p+1)|(n+1).
If it is true then what is the smallest such number? Are such numbers are infinitely many?
I would be extremely grateful for your help.
With best regards,
Nayandeep Deka Baruah Dept. of Math. Sciences, Tezpur University Napaam-784028 Assam, INDIA.
On Wed, 22 Jan 2003, Richard Guy wrote:
Apologies for my wildness. Mike Speciner gives the example n = 8, which generalizes to any odd power of 2. Are these the only examples? Is the sequence
8, 32, 128, 512, 2048, ... (& including any other numbers I've overlooked)
in OEIS ? R.
ID Number: A056729 Sequence: 8,27,32,63,125,128,243,275,343,399,512,567,575,935,1127, 1331,1539,2015,2048,2187,2197,2303,2783,2915,3087,3125,4563, 4913,4991,5103,5719,5831,6399,6859,6875,6929,7055,7139,7625, 8192,8855,12167,12719,14027 Name: If p | n, then p+1 | n+1 for composite n. Comments: The Lucas-Carmichael numbers (A006972) are a subset. Math'ca: Select[ Range[ 2, 10^5 ], ! PrimeQ[ # ] && Union[ Mod[ # + 1, Transpose[ FactorInteger[ # ] ][ [ 1 ] ] + 1 ] ] == {0} & ] See also: Cf. A006972. Keywords: nonn Offset: 1 Author(s): Robert G. Wilson v (RGWv@kspaint.com), Aug 31 2000
On Wed, 22 Jan 2003, Richard Guy wrote:
I'll make a wild guess that it can be proved that no such n exists; I'll copy this to some people who may be able to confirm or deny this.
R.
On Wed, 22 Jan 2003, Mr. Nayandeep Deka Baruah wrote:
Dear Professors Guy and Borwein,
I would like to know from you whether the following result is still a conjecture or has been proved by somebody.
There exists a composite integer n such that for each prime divisor p of n (p+1)|(n+1).
If it is true then what is the smallest such number? Are such numbers are infinitely many?
I would be extremely grateful for your help.
With best regards,
Nayandeep Deka Baruah Dept. of Math. Sciences, Tezpur University Napaam-784028 Assam, INDIA.
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------------------------------------------------------------ W. Edwin Clark, Math Dept, University of South Florida, http://www.math.usf.edu/~eclark/ ------------------------------------------------------------
I want a composite number n having five distinct prime factors (not repeated) such that for each prime factor p of n, p+1|n+1. Nayandeep Deka Baruah TU, INDIA.
On Thu, 23 Jan 2003, Mr. Nayandeep Deka Baruah wrote:
I want a composite number n having five distinct prime factors (not repeated) such that for each prime factor p of n, p+1|n+1.
These are, as you may know, called Lucas-Carmichael numbers. The first such with n prime factors is 588455 (This may be found at http://www.users.globalnet.co.uk/~perry/maths/lucascarmichael/lucascarmichae... Using Maple I found several easily by checking products of 5 distinct primes--just using odd primes up to 100. However I didn't let the program run long before stopping it. Here's one with 6 prime factors also found in a similar manner: 206238815 = (5) (17) (23) (31) (41) (83) --Edwin
On Thu, 23 Jan 2003, Edwin Clark wrote:
On Thu, 23 Jan 2003, Mr. Nayandeep Deka Baruah wrote:
I want a composite number n having five distinct prime factors (not repeated) such that for each prime factor p of n, p+1|n+1.
These are, as you may know, called Lucas-Carmichael numbers. The first such with n prime factors is 588455 (This may be found at
I meant to say "the first such n with 5 prime factors.." ---Edwin
5*7*17*23*43 -----Original Message----- From: math-fun-admin@mailman.xmission.com [mailto:math-fun-admin@mailman.xmission.com]On Behalf Of Mr. Nayandeep Deka Baruah Sent: Thursday, January 23, 2003 00:02 To: Edwin Clark Cc: math-fun@mailman.xmission.com Subject: Re: [math-fun] Re: your mail I want a composite number n having five distinct prime factors (not repeated) such that for each prime factor p of n, p+1|n+1. Nayandeep Deka Baruah TU, INDIA. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Some families of solutions: Suppose n satisfies the property. And suppose that for some p, p+1 divides n+1, while for all prime factors q | n, (q+1) | (p+1)*(p-1). Then n*p*p also satisfies the property: n*p*p + 1 = (n+1)*p*p - (p+1)*(p-1) Example: n = 7 trivially satisfies the property. p = 3 satisfies the above condition. So 7*3*3 also satisfies the property. Now for n = 7*3*3, both 3 and 7 satisfy the above condition, so by repetition, 7^(2j+1) * 3^(2k) satisfies the property. Another example: n = 3*7*19, p = 19.
There exists a composite integer n such that for each prime divisor p of n (p+1)|(n+1).
If it is true then what is the smallest such number? Are such numbers are infinitely many?
I would be extremely grateful for your help.
With best regards,
Nayandeep Deka Baruah Dept. of Math. Sciences, Tezpur University Napaam-784028 Assam, INDIA.
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
a composite integer n such that for each prime divisor p of n, (p+1)|(n+1).
Similarly, Which integers N have these properties? - N has at least two distinct prime factors. - If a prime P divides N, then (P-1)|(N-1) and (P+1)|(N+1). (The first condition just dispels all the boring prime-powers.) Here are the first few N's: 74431 = 7^4 * 31 71528191 = 7^4 * 31^3 178708831 = 7^8 * 31 125780831 = 11^6 * 71 4150390625 = 5^12 * 17 You might enjoy proving: - 2 doesn't divide any such N. - 3 doesn't divide any such N. - The set of such N's is infinite. - If a prime P divides an N, then P==N mod 12. And I wonder: - Is there such an N with more than two distinct prime factors? - Is there such an N congruent to 1 mod 12? (If not, then all N's have an odd number of (multiply-counted) prime factors.) -- Don Reble djr@nk.ca
participants (7)
-
Don Reble -
ed pegg -
Edwin Clark -
Gershon Bialer -
Mike Speciner -
Mr. Nayandeep Deka Baruah -
Richard Guy