June 21st, 2017, 10:18 PM  #1 
Newbie Joined: Jun 2017 From: canada Posts: 9 Thanks: 0  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 
June 22nd, 2017, 02:13 AM  #2 
Global Moderator Joined: Dec 2006 Posts: 18,847 Thanks: 1568 
By definition, z is a divisor on n, so finding z means that n has been factorized.

June 22nd, 2017, 06:20 AM  #3 
Newbie Joined: Jun 2017 From: canada Posts: 9 Thanks: 0 
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. 
June 22nd, 2017, 12:29 PM  #4 
Global Moderator Joined: Dec 2006 Posts: 18,847 Thanks: 1568 
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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