| Title: | Mathematics at DEC |
| Moderator: | RUSURE::EDP |
| Created: | Mon Feb 03 1986 |
| Last Modified: | Fri Jun 06 1997 |
| Last Successful Update: | Fri Jun 06 1997 |
| Number of topics: | 2083 |
| Total number of notes: | 14613 |
Proposed by K. R. S. Sastry, Dodballapur, India.
The infinite arithmetic progression 1+3+5+7+... of odd positive
integers has the property that all of its partial sums
1, 1+3, 1+3+5, 1+3+5+7, ...
are perfect squares. Are there any other infinite arithmetic
progressions, all terms positive integers with no common factor, having
this same property?
| T.R | Title | User | Personal Name | Date | Lines |
|---|---|---|---|---|---|
| 1987.1 | no other solutions | JOBURG::BUCHANAN | Sat Aug 12 1995 04:48 | 18 | |
If the terms of the progression are:
c + ai, i = 0,1,...
The jth partial sum is:
cj + aj(j-1)/2, j=1,2,...
So:
j(4c+2a(j-1)) is a square for all j.
Let j = kl�, where k is square free:
k|4c+2a(kl�-1)
So: k|4c-2a, but k could be any prime. So 4c-2a = 0, and since we are
told that (a,c)=1, the only solution is the one given in the base note.
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