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Topic: Sum of powers of 2 and 3 (Read 513 times) |
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Pietro K.C.
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Re: Sum of powers of 2 and 3
« Reply #1 on: Feb 8th, 2003, 8:53pm » |
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I have a solution that is not very elegant:
The sets
A = {2a(mod 23) : a is natural} and
B = {3b (mod 23) : b is natural}
are equal, because 28 = 3 (mod 23) and 37 = 2 (mod 23). Doing a little scribbling, we come up with:
A = B = {1,2,3,4,6,8,9,12,13,16,18},
and after doing 11 scans over the set we conclude that no pair exists that sums to 23. Hence, there exist no positive integers a,b such that
2a + 3b = 0 (mod 23),
much less equal a power of 23.
I suppose we could improve it thus:
Since A = B, the congruence
2a + 3b = 0 (mod 23)
is equivalent to
2a + 2d = 0 (mod 23).
with 1 <= a,d <= 22 (because of Fermat's little theorem). Supposing a > d, we have:
2a-d + 1 = 0 (mod 23),
and a single glance over A's elements suffices to establish that there is no positive integer k such that
2k = 22 (mod 23).
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