Solving a Simple Diophantine Equation: 3x + 98y = 12

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Homework Help Overview

The discussion revolves around solving a Diophantine equation of the form 3x + 98y = 12, with participants exploring how to handle the negative sign in the equation and the implications of coefficients not being relatively prime. There are also inquiries regarding a separate equation involving squared terms.

Discussion Character

  • Exploratory, Conceptual clarification, Problem interpretation, Assumption checking

Approaches and Questions Raised

  • Participants discuss the transformation of the equation to accommodate the negative coefficient and question the necessity of having relatively prime coefficients for applying the Diophantine procedure. There are also inquiries about the implications of the greatest common divisor (gcd) and how to approach equations involving squared terms.

Discussion Status

Some participants have offered guidance on the relationship between the gcd and the existence of solutions, while others are still seeking clarity on specific methods and concepts related to the squared terms. Multiple interpretations of the problem setup are being explored.

Contextual Notes

Participants are considering the implications of the gcd not being one and the conditions under which solutions exist. There is also mention of constraints related to course content and previous lectures that may influence understanding.

trap101
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find integers x and y such that:

3x - 98y = 12

I understand how to solve for the values, but to make it a diophantine equation I need it to look like 3x + 98y = 12...my question is what do I do with the negative sign? Carry it to the y? i.e: 3x + 98(-y) = 12, then deal with it after I find my values?
 
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trap101 said:
find integers x and y such that:

3x - 98y = 12

I understand how to solve for the values, but to make it a diophantine equation I need it to look like 3x + 98y = 12...my question is what do I do with the negative sign? Carry it to the y? i.e: 3x + 98(-y) = 12, then deal with it after I find my values?

If you want positive coefficients, sure, do it that way.
 
Thanks.

I had two other questions. One was quick the other not so much: Back to this equation above, I was wondering since the equation is 3x + 98(-y) = 12, how can I use the diophantine procedure if the values are not relatively prime? i.e don't I have to set it up in the form:

3x + 98(-y) = 1...and then once I plow through with the euclidean algorithm Multiply through by 12 in this case. What happens if I don't get a gcd of 1 between my two values? Can I still use this?The other question had to do with squared terms:

13x^2 - 23y^2 = 1...how do I deal with the squares?
 
trap101 said:
Thanks.

I had two other questions. One was quick the other not so much: Back to this equation above, I was wondering since the equation is 3x + 98(-y) = 12, how can I use the diophantine procedure if the values are not relatively prime? i.e don't I have to set it up in the form:

3x + 98(-y) = 1...and then once I plow through with the euclidean algorithm Multiply through by 12 in this case. What happens if I don't get a gcd of 1 between my two values? Can I still use this?The other question had to do with squared terms:

13x^2 - 23y^2 = 1...how do I deal with the squares?

If the gcd isn't one, then you won't have a solution at all the if constant on the right side isn't divisible by the gcd. If it is, then divide the equation by the gcd. The second one I'd need to give some more thought to. Do you have any ideas from the course?
 
Dick said:
If the gcd isn't one, then you won't have a solution at all the if constant on the right side isn't divisible by the gcd. If it is, then divide the equation by the gcd. The second one I'd need to give some more thought to. Do you have any ideas from the course?

I don't remember him talking about it specifically this time, but I sat in on some lectures during the last semester and what I remember is something along the lines of some sort of factoring technique...vague I know. I'm stumped and haven't found much online either.
 
How about considering ##13x^2 \equiv 1\,mod\,23?##?
 

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