[seqfan] Re: (no subject)
franktaw at netscape.net
franktaw at netscape.net
Mon Nov 9 17:22:27 CET 2009
There was a discussion here just a couple of weeks ago about what
numbers can be added to a triangular number to get a square.
Specifically, for the first few values, look at the numbers mod 9. The
triangular numbers mod 9 are 0, 1, 3, and 6; the squares mod 9 are 0,
1, 4, and 7. If you look at the differences, you will see that 2 and 5
cannot be obtained. This takes care of 2, 5, 11, 16, 20, 25, ....
In general, to determine if a number k can be represented in this way,
factor 8k-1. If any prime p == 3 or 5 (mod 8) appears as a factor an
odd number of times (i.e., p, p^3, p^5, ...; but p^2, p^4, p^6, ... are
OK), k cannot be represented as the difference of a square and a
triangular number. Otherwise it can be.
When a number can be so represented, there are infinitely many
solutions. Take one large enough, and there will be no other squares
between n(n+1)/2 and n(n+1)/2 + k, so restricting the square to the
next one, as you are doing, doesn't change which numbers can be
obtained.
What is happening is that we can write n(n+1)/2 + k = m^2. Multiply by
8 and rearrange algebraically, and you get:
2*(2m)^2 - (2n+1)^2 = 8k-1.
This is a generalized Pell equation, a subject that is well understood:
see for example
http://mathworld.wolfram.com/PellEquation.html.
Franklin T. Adams-Watters
-----Original Message-----
From: c.zizka at email.cz
Dear seqfans,
let a(n) is the difference between n-th triangular number and next
perfect
square (submitted to OEIS today).
a(n)= (ceiling(n*(n+1)/2))^2 - n*(n+1)/2
Interestingly a(n) takes values
{0,1,3,4,6,8,9,10,13,15,16,19,21,22,...} and it
seems no a(n) is from {2,5,7,11,12,14,17,18,20,23,...} .
In OEIS we can found for a(n)=0 A001108, for a(n)=1 A006451, for
a(n)=3
A154138, for a(n)=4 A154139 , for a(n)=6 A154140, for a(n)=8
A154141, for
a(n)=9 A154142, for a(n)=10 A154143.
Why can not be :
(ceiling(n*(n+1)/2))^2 = 2+ n*(n+1)/2
:-)
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