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Brain teaser
Posted by davelindell on February 23, 2021 at 10:38 pmWhat is the smallest positive integer, y, that makes 1729y?ý+1 equal a perfect square?
johnnolton replied 3 years, 4 months ago 13 Members · 72 Replies 
72 Replies

Posted by: @mathteacher
Is there a way to solve this other than trying each value? Properties of integers is not one of my strong suits.
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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.

Posted by: @holycow
Nope. It works to the precision of my calculator, but the expression yields 15087454565 and 122831^2 is 15087454561.
You can see that it doesn’t work just by keeping only last digits in your head as you work through it, and not worrying about the rest of the big product digits.
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After additional research I found that
1729???ú??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.

@holycow Nope, I get 122831.00001628253453824718136713 which is well within the accuracy of my crappy 64 bit calculator so y is not = 2954.
Plenty good enough for setting a monument though and I’d be miffed if you pincushioned my mon 0.00001′ next to it.


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.

I get the same number as the MathTeacher “35″
JOHN NOLTON

If y=0 isnt ok then
y=1729*13! which is the integer 10766518963200
The side of the square is then 447685271124001
not so small ????

Not quite. That number is actually 447685271124000.969078681, truncated at 9 decimal places.
This site may have enough capacity to find the answer, or it may not: https://www.mathsisfun.com/calculatorprecision.html

Not mine, either. I’d be a bad choice for a crypto job.


The square root of 2^668122 is just 2^334061; take half the exponent. But, we want 1729*y^2+1 to be a perfect square and your y produces a prime number for that expression, which can’t be a perfect square.
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