An old puzzle about two numbers

[XYYXF]

Originally Posted by Kees

I am missing something, but do not know what

You are missing 53 as a possible sum

41*12 = 82*6

[Kees]

Hmm
accept solution


Kees

[philmoore]

Originally Posted by XYYXF

So there's the solution (spoiled). :)

Mersenne knows than Fermat can't find A and B, i.e. that A*B is non-simple. But Mersenne was informed only about A+B. It means that every product n*(A+B-n), where both n and A+B-n are less than 100 and greater than 1, is non-simple.

I'm having trouble with this solution, because of the following:

Mersenne would know the solution if there is only one non-simple product among the n*(A+B-n), so Mersenne only knows that there must be at least two non-simple solutions among the products n*(A+B-n), not that all of these products must be non-simple.

[XYYXF]

philmoore, if there is at least one simple product, then, from Mersenne's point of view, there will be a chance that namely this product was reported to Fermat, so Fermat knows A and B.

[mfgoode]

Originally Posted by akruppa

So please tell us what the values xy and x+y are.

Alex

Of the many answers up to 100 take arbitrary values for x and y keeping in mind that x + y should be even and the discriminant a +ve square and insert in the given eqn.

Thus x = 4 +- 1 i.e. 5 or 3 and correspondingly y =3 or 5

Another is x = 7 or 3 and y = 3 or 7 resp.

The eqn suits negative values also.

Take x = 3 and y = -5 or x = - 5 y= 3.

I have not pursued further for both -ve values . I presume it can be done.

Mally

[akruppa]

We have already established that x+y are not even. Read Fermat's and Mersenne's conversation again. Also, it is sufficient that the discriminant is an integer square for the solutions to be integers.

You have just shown that your equation can be used to find x and y, given x+y and xy. We already know that. You still haven't said anything about how to find x+y and xy from Fermat's and Mersenne's conversation. Not that it matters any more, as XYYXF has posted the solution by now.

Alex

[philmoore]

I see, I overlooked the "Of course you know" prefacing Fermat's first statement, and was making the problem harder!