I have two multigrades. They need fixing. In each, one of the numbers is off.

$1^n + 11^n + 13^n + 33^n + 35^n + 45^n =
3^n + 5^n + 21^n + 25^n + 41^n + 45^n$
; n = 1,2,3,4,5

The second is:

$1^k + 2^k + 31^k + 32^k + 55^k + 61^k + 68^k =
17^k + 20^k + 23^k + 44^k + 49^k + 64^k + 67^k$
; k = 2,4,6,8,10

There's a little story behind these. I expect someone to fix 'em by today.

2. Originally Posted by wonderboy1953
I have two multigrades. They need fixing. In each, one of the numbers is off.

$1^n + 11^n + 13^n + 33^n + 35^n + 45^n =
3^n + 5^n + 21^n + 25^n + 41^n + 45^n$
; n = 1,2,3,4,5

The second is:

$1^k + 2^k + 31^k + 32^k + 55^k + 61^k + 68^k =
17^k + 20^k + 23^k + 44^k + 49^k + 64^k + 67^k$
; k = 2,4,6,8,10

There's a little story behind these. I expect someone to fix 'em by today.
I don't think I understand what you're saying.
Is it that, in 'multigrade 1' for example, that you can change one of the numbers 1,11,13,33,35,45,3,5,21,25,41 or 45, for a different number and the equality holds?

Say for n=1, the difference between RHS and LHS is 2, so any of the numbers 'are off' by 2, so adding 2 onto any of the numbers 'fixes' it. Is that what you mean? Because if it is, there is no one number you can change in the first equality that will make it true when n=2,3,4 or 5.

Strictly speaking neither multigrade is an equality before the fixing; potentially an equality after the fixing.

"so adding 2 onto any of the numbers 'fixes' it." No, just one of the numbers can fix the multigrade so you may add (or subtract) 2 with one of the numbers in the first multigrade which you have to figure out which one.

4. Originally Posted by wonderboy1953
Strictly speaking neither multigrade is an equality before the fixing; potentially an equality after the fixing.

"so adding 2 onto any of the numbers 'fixes' it." No, just one of the numbers can fix the multigrade so you may add (or subtract) 2 with one of the numbers in the first multigrade which you have to figure out which one.
Oh OK, i get it now, thanks. In that case the first multigrade is fixed by changing the second 45 to 43.

What's the story behind them?

I'm waiting on the second multigrade fix, then I'll do the story.

6. Originally Posted by wonderboy1953
I'm waiting on the second multigrade fix, then I'll do the story.
Change the 2 to a 28.

7. ## To pomp

You got 'em both.

The story is these multigrades come from a book on number theory. I don't believe the author made those errors, rather they come from the publisher.

I've ran across several math books with errors in them. It occurred to me that puzzles can be based on those errors which can be interesting and profitable.