# Conjecture about the set of Sphenic numbers

Sum of a set of sphenic numbers can't be equal to the sum of any other set of sphenic numbers.

By that I meant, Say S is the set of sphenic numbers. Let S$_1$ $\subset$ S. Then there is no such S$_2$ $\subset$ S so that,

S$_1$ $\neq$ S$_2$ AND $\sum S_1$ $=$ $\sum S_2$

Question 1 : Is this statement correct?

Question 2 : Is this formulation mathematically right? I mean even if the statement is wrong, is the way that I expressed it conveys mathematical notations/rules etc. Or, how a mathematician would write it if s/he intended to convey the same message?

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1) The statement is not correct. 230+1310=231+1309. Sphenic numbers gives the first sequential pair (and implies there are more) and the first run of three.

2) I'm not a mathematician, but the formulation seems fine to me. You might define Sphenic numbers so people don't have to look it up

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Thanks, you're right. –  user1869 Oct 14 '10 at 14:13
I believe he is looking for same sums, not necessarily consecutive. For instance 2*3*5 + 11*3*5 = 13*3*5 is a smaller set than what you have. –  Aryabhata Oct 14 '10 at 14:42
This was just the first example I found. Also, pick any pairs of primes with a common difference, like 7/13 and 23/29. Then pq*(7+29)=pq*(13+23) for any p,q that are not 7, 13, 23, or 29. –  Ross Millikan Oct 14 '10 at 15:21
@Ross: Yes, I was only pointing out that 'first pair' is misleading. –  Aryabhata Oct 14 '10 at 16:15
@Moron: your version also gives an infinite set from any pair of twin primes. –  Ross Millikan Oct 14 '10 at 16:22

Question 2: Instead of $\sum S_1$ you can write $$\sum_{k \in S_1} k.$$

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Thanks. Is it for the reason that in the literature k[small letter] denotes elements and S[capital letter] denotes sets? Or, for other reasons? –  user1869 Oct 14 '10 at 17:26
@Sazzad: Upper or lower case is not the main point (although the convention is as you describe). Rather it's that what comes after the summation sign should describe what you are adding up, in your case numbers (not sets). –  Hans Lundmark Oct 14 '10 at 17:49
I see. Thank you very much. –  user1869 Oct 14 '10 at 18:25