Is there any truth to this?

Discussion in 'Number Theory' started by PixelEight, Apr 18, 2021.

  1. PixelEight

    PixelEight

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    Say we want to factorize 87 .. we know that 87=3*29
    We know that there exists two perfect squares that are separated by 87 units. These two squares are 169 and 256.
    256-169=87. And if we take the difference of square roots of the two perfect squares (and addition) we get the prime factors
    of 87. eg. sqrt(256)-sqrt(169)=3 ... sqrt(256)+sqrt(169)=29

    So the problem remains of finding these two perfect squares.

    13 21 23 25 27 29 31
    87 100 121 144 169 196 225 256
    34 57 82 109 138 169

    5^2+9 6^2+21 7^2+33 8^2+45 9^2+57 10^2+69

    12n+9+(n+5)^2 = n^2+22n+34

    (11+2n)((floor(sqrt((9+12n)mod(11+2n)))+1))+(floor(sqrt((9+12n)mod(11+2n)))+1)*((floor(sqrt((9+12n)mod(11+2n)))+1)-1)=9+12n

    According to WolframAlpha: n=5

    so: 5^2+22*5+34=169

    sqrt(169+87) - sqrt(169) = 3
    sqrt(169+87) + sqrt(169) = 29
     
    PixelEight, Apr 18, 2021
    #1
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