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Divisibility of N (Posted on 2013-06-15) Difficulty: 3 of 5
The problem Divisibility of 29 essentially asks whether three 4th powers can sum to a multiple of 29 if they are not all multiples of 29.

The entry for 29 in The Penguin Dictionary of Curious and Interesting Numbers by David Wells (1987) contains:

No sum of three 4th powers is divisible by either 5 or 29 unless they all are. [Euler]

1. If three aren't enough, how many 4th powers does it take to be divisible by either 5 or 29 where they aren't all?

2. If possible, find the next number beyond 5 and 29 that does not divide a sum of three 4th powers.

3. Prove every even number takes at most two 4th powers.
For example using 18 we have 34+154 = 50706 = 18*2817

4. What is the largest number of 5th powers whose sum is divisible by a number N where they aren't all divisible by N?

5. Prove that for higher powers, there is no limit to how many numbers it can take.

No Solution Yet Submitted by Jer    
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Question No Subject | Comment 1 of 4
Prove every even number takes at most two 4th powers.

every even number ??
  2, 4,    100?

please explain:  for every  even divisor 2k there exist  a number 2mk , equal to a^2+b^2, none of those addends divisible by 2k .

Is this the thing to be proven??

Edited on June 15, 2013, 2:38 pm
  Posted by Ady TZIDON on 2013-06-15 14:14:25

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