Find the possible nonnegative integer values of
X so that:
corresponds to a nonnegative integer power of 2.
Prove that no other value of X conforms to the given conditions.
Note: Adapted from a problem appearing at Spanish Mathematical Olympiad in 1986.
It looks like x = {0,1,3} equate to 2^{2,3,7}
We can show that the expression 5^x + 3 is always
divisible by 4:
Since 5 = 4+1, expand (t+1)^x, where t is 4 ...
(4+1)^x will always be a series of terms with 4 as a factor and then one more term of 1, so this is always 1 mod 4. Adding 3 makes it 0 mod 4.
Maybe playing with the Pascal's Triangle coefficients and the exponents might show a pattern
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Posted by Larry
on 2023-06-18 10:05:15 |
an observation
