[math-fun] Sums of two cubes
I am sure I am very late to this party, but I have been looking at positive sums of two (not necessarily positive) cubes, catalogued on OEIS as A045980. The corresponding set for squares is easily characterized in terms of prime factorizations: the exponents of primes of the form 4k-1 must be even. (I hope I have that characterization right; if it's not, it's close.) The experience for sums of two squares inspires me to factorize the first few dozen entries in A045980. The result is very suggestive, about as suggestive as it's possible to be with actually, you know, *suggesting* anything. The most striking feature is a marked preference for primes of the form 6k+1. Of course if a number n is the sum of two cubes, then nm^3 is also, for all integers m. But the converse is not true. For example, 1^3 + 7^3 = 344 = 2^3*43, but 43 is not the sum of two cubes. The property is not multiplicative: for example, 2 and 7 are sums of two cubes, but 14 isn't. That having been said, it displays frustrating *hints* of multiplicativity. After staring at these numbers for a couple of days, I have come up with a hard question related to this spooky para-multiplicativity. For any k, let Ck be the set of integers n such that kn is the sum of two squares. C1, for instance, is A045980 itself. C2 is {1, 4, 8, 13, 14 ...}, which is not in OEIS. (In fact, none of the Ck's I looked at were in OEIS except C1.) Here is my question: do there exist *any* two integers j and k such that Cj = Ck?
Allan, last month you said: After staring at these numbers for a couple of days, I have come up with a hard question related to this spooky para-multiplicativity. For any k, let Ck be the set of integers n such that kn is the sum of two squares. C1, for instance, is A045980 itself. C2 is {1, 4, 8, 13, 14 ...}, which is not in OEIS. (In fact, none of the Ck's I looked at were in OEIS except C1.) Here is my question: do there exist *any* two integers j and k such that Cj = Ck? Me: sorry for the delay! I added three of your C functions to the OEIS - see A222304-A222306. One observation is that C_3 = 3*C_9. Neil On Sat, Jan 26, 2013 at 3:18 PM, Allan Wechsler <acwacw@gmail.com> wrote:
I am sure I am very late to this party, but I have been looking at positive sums of two (not necessarily positive) cubes, catalogued on OEIS as A045980. The corresponding set for squares is easily characterized in terms of prime factorizations: the exponents of primes of the form 4k-1 must be even. (I hope I have that characterization right; if it's not, it's close.)
The experience for sums of two squares inspires me to factorize the first few dozen entries in A045980. The result is very suggestive, about as suggestive as it's possible to be with actually, you know, *suggesting* anything. The most striking feature is a marked preference for primes of the form 6k+1. Of course if a number n is the sum of two cubes, then nm^3 is also, for all integers m. But the converse is not true. For example, 1^3 + 7^3 = 344 = 2^3*43, but 43 is not the sum of two cubes.
The property is not multiplicative: for example, 2 and 7 are sums of two cubes, but 14 isn't. That having been said, it displays frustrating *hints* of multiplicativity.
After staring at these numbers for a couple of days, I have come up with a hard question related to this spooky para-multiplicativity. For any k, let Ck be the set of integers n such that kn is the sum of two squares. C1, for instance, is A045980 itself. C2 is {1, 4, 8, 13, 14 ...}, which is not in OEIS. (In fact, none of the Ck's I looked at were in OEIS except C1.) Here is my question: do there exist *any* two integers j and k such that Cj = Ck? _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Dear Friends, I have now retired from AT&T. New coordinates: Neil J. A. Sloane, President, OEIS Foundation 11 South Adelaide Avenue, Highland Park, NJ 08904, USA Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com
-----Original Message----- From: math-fun-bounces@mailman.xmission.com [mailto:math-fun- bounces@mailman.xmission.com] On Behalf Of Neil Sloane Sent: Friday, February 15, 2013 2:05 PM To: math-fun Subject: Re: [math-fun] Sums of two cubes
Allan, last month you said:
After staring at these numbers for a couple of days, I have come up with a hard question related to this spooky para-multiplicativity. For any k, let Ck be the set of integers n such that kn is the sum of two squares. C1, for instance, is A045980 itself. C2 is {1, 4, 8, 13, 14 ...}, which is not in OEIS. (In fact, none of the Ck's I looked at were in OEIS except C1.) Here is my question: do there exist *any* two integers j and k such that Cj = Ck?
[David Wilson] I presume you mean distinct integers j, k >= 1. Define f(j, k) to be the smallest integer in exactly one of Cj and Ck. If Cj = Ck, f(j, k) is undefined. Empirically (testing j, k <= 300), it looks as if f(j, k) is always defined and appears to satisfy f(j, k) <= (j+k)^2/4.
participants (3)
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Allan Wechsler -
David Wilson -
Neil Sloane