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a method/algorithm to solve a nonlinear Diophantine equation like this

Simpler Way to Solve This Diophantine EquationSolve this non-linear diophantine equation?Amount of solutions to the Diophantine equation of FrobeniusAlternative method to solve quadratic Diophantine equationsFermat's factorization algorithmConvert a Pair of Integers to a Integer, Optimally?Three Variable Nonlinear Diophantine EquationAlgorithm for diophantine equationSemiprime factorizationInteger Factoring - Check if $N=a^b$ for some integers $a$ and $b$

0

$begingroup$

I have a Diophantine equation like this

10xy + 7y + 9x = 123456789..... (a 270 digit number!!)

I have tried an online calculator to solve it
https://www.alpertron.com.ar/QUAD.HTM,
but it is factoring that huge number, so it is taking a very very long time.

My question is :
Is there any algorithm to solve Diophantine equations on this specific form

10xy + 7y + 9x = c 

where c is a huge ~300 digits number, without the need for factorization?

I need a solution that doesn't take forever to calculate,

the only constraint is that x and y are integers

thanks.

algorithms diophantine-equations

share|cite|improve this question

edited 18 hours ago

Omar

asked Mar 3 at 23:04

OmarOmar

133

New contributor

Omar is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  why do you want this at all? What is the source of the problem?
  $endgroup$
  – Will Jagy
  Mar 3 at 23:11
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  $(10x+7)(10y+9) = 10c + 63.$ This is still an enormous number, but might be easy to factor.
  $endgroup$
  – Will Jagy
  Mar 3 at 23:14
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @Will Jagy but I still need to know the factors of 10c+63 , right? that factorization is going to take a long time
  $endgroup$
  – Omar
  Mar 3 at 23:21
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  You do need to understand that any $(x,y)$ solution gives a factoring of $10c+63$ into two numbers. The collection of ALL $(x,y)$ solution gives a complete factorization.
  $endgroup$
  – Will Jagy
  Mar 3 at 23:26
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  yes I understand you, but it is the same problem
  $endgroup$
  – Omar
  Mar 3 at 23:42
  

  
  
  
  

 | 
show 7 more comments

0

$begingroup$

I have a Diophantine equation like this

10xy + 7y + 9x = 123456789..... (a 270 digit number!!)

I have tried an online calculator to solve it
https://www.alpertron.com.ar/QUAD.HTM,
but it is factoring that huge number, so it is taking a very very long time.

My question is :
Is there any algorithm to solve Diophantine equations on this specific form

10xy + 7y + 9x = c 

where c is a huge ~300 digits number, without the need for factorization?

I need a solution that doesn't take forever to calculate,

the only constraint is that x and y are integers

thanks.

algorithms diophantine-equations

share|cite|improve this question

edited 18 hours ago

Omar

asked Mar 3 at 23:04

OmarOmar

133

New contributor

Omar is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  why do you want this at all? What is the source of the problem?
  $endgroup$
  – Will Jagy
  Mar 3 at 23:11
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  $(10x+7)(10y+9) = 10c + 63.$ This is still an enormous number, but might be easy to factor.
  $endgroup$
  – Will Jagy
  Mar 3 at 23:14
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @Will Jagy but I still need to know the factors of 10c+63 , right? that factorization is going to take a long time
  $endgroup$
  – Omar
  Mar 3 at 23:21
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  You do need to understand that any $(x,y)$ solution gives a factoring of $10c+63$ into two numbers. The collection of ALL $(x,y)$ solution gives a complete factorization.
  $endgroup$
  – Will Jagy
  Mar 3 at 23:26
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  yes I understand you, but it is the same problem
  $endgroup$
  – Omar
  Mar 3 at 23:42
  

  
  
  
  

 | 
show 7 more comments

$begingroup$

I have a Diophantine equation like this

10xy + 7y + 9x = 123456789..... (a 270 digit number!!)

I have tried an online calculator to solve it
https://www.alpertron.com.ar/QUAD.HTM,
but it is factoring that huge number, so it is taking a very very long time.

My question is :
Is there any algorithm to solve Diophantine equations on this specific form

10xy + 7y + 9x = c 

where c is a huge ~300 digits number, without the need for factorization?

I need a solution that doesn't take forever to calculate,

the only constraint is that x and y are integers

thanks.

algorithms diophantine-equations

share|cite|improve this question

edited 18 hours ago

Omar

asked Mar 3 at 23:04

OmarOmar

133

New contributor

Omar is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

$endgroup$

10xy + 7y + 9x = 123456789..... (a 270 digit number!!)

10xy + 7y + 9x = c 

share|cite|improve this question

edited 18 hours ago

Omar

asked Mar 3 at 23:04

OmarOmar

133

New contributor

Omar is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

share|cite|improve this question

edited 18 hours ago

Omar

asked Mar 3 at 23:04

OmarOmar

133

New contributor

Omar is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

asked Mar 3 at 23:04

OmarOmar

133

New contributor

Omar is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

asked Mar 3 at 23:04

OmarOmar

133