# Is there any truth to this?

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

1. ### PixelEight

Joined:
Apr 18, 2021
Messages:
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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

2. ### PixelEight

Joined:
Apr 18, 2021
Messages:
2
0
Repost:

Integer Factorization [email protected]
===============

Say we want to factorize 87 .. 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

(2n+11)*Quotient[(12n+9),(2n+11)]+Quotient[(12n+9),(2n+11)]*(Quotient[(12n+9),(2n+11)]-1)=12n+9

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, Jun 8, 2021

3. ### paulejking

Joined:
Jul 26, 2021
Messages:
4
0
I see a lot of math here, but you never stated the overall problem.... something, something, prime factors which have a property, something something. Not really good enough.

"So the problem remains of finding these two perfect squares."​

"These two"? You then proceed with a big list of numbers (more than the two you claim) with a ton of perfect squares, followed by a bunch of steps you didn't explain and don't seem to have a rhyme or reason. Then I see the numbers 3 and 29 on the last step. So what? What is your point?

paulejking, Jul 26, 2021