What is the smallest positive integer, y, that makes 1729y²+1 equal a perfect square?

35

That is a very difficult problem and forms the basis of some Internet encryption. The solution could be in the tens of billons if y > (a big number). I poked around and could find no easy way to solve it. I suspect your problem is a teaser and is unsolvable without big iron supercomputers.

If someone pops up with the solution I'll be mightily impressed.

After additional research I found that

1729 · 2^{668122} + 1 |

is prime so the problem reduces to y2 = 2^ 668122 to solve for y which is no trivial task. I'm convinced it's an even number though because of the +1 factor.

The answer is obviously Pi. Pi is the answer to almost all things. If not, it's Avogadro's number: 6.02 x 10 to the 23rd.

If I ever need to solve the problem the OP presented as a matter of land surveying, I shall turn in my license. I am confident my license is not at risk.