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Topic: cotpi 46 - Raised to the same power
Replies: 12   Last Post: Apr 10, 2012 4:26 PM

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 Mark Brader Posts: 87 Registered: 12/10/04
Re: cotpi 46 - Raised to the same power
Posted: Apr 8, 2012 11:27 PM

>>>> Two given positive integers are raised to the same power and
>>>> added. The sum is a power of 2. What is the maximum possible
>>>> difference between the two given integers?

>>> There is no maximum.

>> And even if we exclude ^1 as trivial, then there is still no limit to the
>> difference, for if:
>> a^k + b^k = 2^n
>> is a solution, then
>> (2a)^k + (2b)^k = 2^(n+k)
>> is another.

> Yes, we will have to exclude ^1 since there was
> an error in the problem statement. Sorry about that.

(Smiles)

> Your argument works only if there is some a and b
> such that |a - b| > 0.

Okay, so when the question didn't say that the two integers had to
be distinct, that *was* deliberate. Then we have two possible cases.
Either (1) with the restriction that k>1, there is no solution with
a != b, so the maximum is 0; or (2) an example exists with a != b,
and there is no maximum.

Some brute-force searching doesn't turn up any solutions, so I'm
guessing that maximum is 0.
--
Mark Brader | "It is only a guess, of course.
msb@vex.net | I hope none of you ever finds out for certain."
Toronto | -- Insp. Grandpierre (Peter Stone, "Charade")

Date Subject Author
4/8/12 cotpi
4/8/12 Prai Jei
4/8/12 cotpi