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        riddles >> easy >> Sum of powers of 2 and 3
        
(Message started by: NickH on Feb 8^th, 2003, 3:39am)

Title: Sum of powers of 2 and 3
Post by NickH on Feb 8^th, 2003, 3:39am

Show that 2^a + 3^b = 23^c has no solution in positive integers.

Title: Re: Sum of powers of 2 and 3
Post by Pietro K.C. on Feb 8^th, 2003, 8:53pm

I have a solution that is not very elegant:

The sets

A = {2^a(mod 23) : a is natural} and
B = {3^b (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

2^a + 3^b = 0 (mod 23),

much less equal a power of 23.

I suppose we could improve it thus:

Since A = B, the congruence

2^a + 3^b = 0 (mod 23)

is equivalent to

2^a + 2^d = 0 (mod 23).

with 1 <= a,d <= 22 (because of Fermat's little theorem). Supposing a > d, we have:

2^a-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

2^k = 22 (mod 23).