Integer solutions for equations

  • Context: High School 
  • Thread starter Thread starter 69911e
  • Start date Start date
  • Tags Tags
    Integer
Click For Summary

Discussion Overview

The discussion revolves around finding integer solutions for equations involving positive integers, specifically focusing on the equation 30x + 19 = 7y + 1 = a. Participants explore methods to derive a universal formula for a specific constant C5 in a generalized equation involving positive integers.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested, Mathematical reasoning

Main Points Raised

  • One participant seeks to understand how to find integer solutions when parameters are limited to positive integers, providing a specific example and asking for a formula for C5.
  • Another participant references the Chinese remainder theorem as a basis for the existence of solutions and suggests that the equation can be interpreted in modular arithmetic.
  • A participant mentions that coefficients C1 and C3 are co-prime, which may affect the approach to finding solutions.
  • There is mention of an algorithm to find solutions, though it is noted that it may not yield a straightforward closed formula.
  • One participant expresses optimism about the algorithm's complexity, suggesting it may be manageable if limited to two equations.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the existence of a closed formula for C5, and multiple approaches and interpretations are presented without resolution.

Contextual Notes

Participants acknowledge the need for co-primality of coefficients and the limitations of the algorithm discussed, indicating that the complexity may vary based on the number of equations involved.

69911e
Messages
30
Reaction score
1
I am trying to understand how to find solutions for a problem when parameters are limited to positive integers.
Example:
30x+19= 7y+1 =a ; where x,y,a are positive integers

Wolframalpha outputs:
a = 210 n + 169, x = 7 n + 5, y = 30 n + 24, n element Z(integers)

30*7= 210 (obviously)
How do I calculate 169? I can do it by hand, but am likely missing something obvious generating a universal equation.

I am trying to find a formula to solve for C5 below:
(C1 * X) +C2 = (C3*Y) + C4 = (C1*C3*n) +C5 ; where C1,C2,C3,C4 are constants and X,Y,n are the set of positive integers. Looking for a formula for smallest integer solution C5

Any suggestions?
 
Mathematics news on Phys.org
The existence is a result of the Chinese remainder theorem, and with the formulas there you can also construct the smallest solution.

##30x+19=a## can be interpreted as ##a=19 \pmod{30}##.
 
MFB: Thanks for the link.
I forgot to specify, coefficients (C1&C3) of x and y are also defined as co-prime.

This is a small part of a (non-school) prime number related summer project I am working on with my 13&16 year old and we trying no to get bogged down on this equation. If it does not a have a formula solution for C5 (above), we may need to move onto a different path.
 
There is an algorithm to find it. It is not directly a nice closed formula.
 
MFB: It seems not too complex of an algorithm if limited to 2 equations. We may be able to use it and define the bounds needed.
Thanks
 

Similar threads

MHB Positive Integer Solutions of $(x^2+y^2)^n=(xy)^{2014}$
  • · Replies 2 ·
Replies
2
Views
2K
High School How to find an integer solution to a nonlinear equation?
  • · Replies 6 ·
Replies
6
Views
2K
MHB Integer solutions of system of equations
  • · Replies 5 ·
Replies
5
Views
2K
MHB Direct solution for two unknowns in two equations
  • · Replies 2 ·
Replies
2
Views
2K
High School A symmetric problem has an asymmetric solution - why?
  • · Replies 21 ·
Replies
21
Views
2K
High School On the representation of integers?
  • · Replies 16 ·
Replies
16
Views
3K
Graduate Find positive integer solutions to a/(b+c)+b/(a+c)+c/(a+b)=4
  • · Replies 1 ·
Replies
1
Views
30K
High School An odd integer series formula?
  • · Replies 5 ·
Replies
5
Views
2K
High School Is there a way to find the integer representation of a real number?
  • · Replies 4 ·
Replies
4
Views
2K
Find integer points on this equation
  • · Replies 8 ·
Replies
8
Views
2K