[seqfan] Re: p = (k + 1)^2 - k = (m + 1)^3 - m and p = (k + 1)^2 + k = (m + 1)^3 + m

[Andrew N W Hone]

These correspond to integer points (k,n) on the elliptic curve 

n^2 = k^3 -16*k + 16

which has j-invariant 110592/37. Presumably these are the only such points (up to y->-y), but 
I have not checked. (Siegel's theorem says that there are only finitely many such points: see 
http://math.stackexchange.com/questions/32847/integral-points-on-an-elliptic-curve for an 
informal discussion.)  

I recognize this curve from its j-invariant: it is the same as the curve corresponding to the 
Somos-4 sequence A006720. 

However, I'm afraid I missed the discussion leading up to this. I'm not sure whether one should put 
this (finite) sequence on the OEIS. One could construct infinitely many sequences of this type by 
picking particular Weierstrass models of particular elliptic curves, so is there a strong reason to 
include this one? 

All the best,
Andy 
 

________________________________________
From: SeqFan [seqfan-bounces at list.seqfan.eu] on behalf of юрий герасимов [2stepan at rambler.ru]
Sent: 26 September 2013 09:00
To: seqfan at list.seqfan.eu
Subject: [seqfan] Re: p = (k + 1)^2 - k = (m + 1)^3 - m and p = (k + 1)^2 +     k = (m + 1)^3 + m

Dear SeqFans, tranks for the help.

However in OEIS I have not found the name of sequence "numbers n that are sqrt(k^3 - 16*k + 16) for some k" with the membres: 0, 1, 4, 8, 24.

Offer or no offer it in OEIS?

Best regards,

JSG

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