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June 21st, 2017, 10:18 PM   #1
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2 equations , 3 variables

Sorry as I'm posting this for the second time....
The first time, it was in the wrong section.

Here is my problem, I have two equations:

a) n=yz
b) n=6xy+y thus n=(6x+1)y thus z=6x+1

n=753779

I've learned trough my journey that it's a Diophantine equation because x, y and z need to be positive integers.
Also, that 6xy+y is a special equation of axy+bx+cy+d=0 where b=0, though I'm not certain.
And that I may have to look for direction in affine geometry. It seems very harsh, so I'm turning to the community before attacking this matter...
thank you very much.

Is there a way to find solutions WITHOUT factoring ?
Can you tell me more about these equations for my future researches?
joswhale

Last edited by skipjack; June 22nd, 2017 at 02:15 AM. Reason: VERY Curious
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June 22nd, 2017, 02:13 AM   #2
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By definition, z is a divisor on n, so finding z means that n has been factorized.
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June 22nd, 2017, 06:20 AM   #3
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for any number there will always be the z=1 and y =n. Is it possible to find a mathematical path to find other solutions without factorization.
As i don't want to go through the spectrum of prime numbers to find a solutions. The numbers of trial will be very high for a prime factor...

thank you.

Last edited by joswhale; June 22nd, 2017 at 06:40 AM.
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June 22nd, 2017, 12:29 PM   #4
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You won't have z = 1, as that would mean that x = 0, which isn't positive. You can't avoid factorizing n to find solutions with y > 1, because you define n as equal to yz.
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