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# Finding multiple solutions to Diophantine equations watch

1. Suppose I need to find integer solutions to . I know that there will also be solutions for every such that .

Suppose I already know the values of and . Is there a quick way to find values for and without having to solve another equation?
2. (Original post by MR1999)
Suppose I need to find integer solutions to . I know that there will also be solutions for every such that .

Suppose I already know the values of and . Is there a quick way to find values for and without having to solve another equation?
I'm puzzled, don't you just read solutions off from ? I'm assuming that as you've called this a Diophantine problem, A and B are integers.
3. (Original post by MR1999)
Suppose I need to find integer solutions to . I know that there will also be solutions for every such that .

Suppose I already know the values of and . Is there a quick way to find values for and without having to solve another equation?
(Original post by Gregorius)
I'm puzzled, don't you just read solutions off from ? I'm assuming that as you've called this a Diophantine problem, A and B are integers.
I think the question has been badly paraphrased (in particular, what has 14 got to do with the problem as stated...?)

But I think what the OP actually wants to know is (with me giving concrete examples):

Suppose we have a simple Diophantine equation such as 18x+15y = 15.

It's obvious that x = 0, y = 1 is a solution, but the general form x = 5n, y=1-6n is less obvious.

But (to the OP), it's still pretty straightforward: it's clear that 18(15) + 15(-18) = 0, (which would give x = 15n, y = 1-18n). For full generality we divide by the highest common factor of (18, 15), which is 3.
4. (Original post by DFranklin)
I think the question has been badly paraphrased (in particular, what has 14 got to do with the problem as stated...?)

But I think what the OP actually wants to know is (with me giving concrete examples):

Suppose we have a simple Diophantine equation such as 18x+15y = 15.

It's obvious that x = 0, y = 1 is a solution, but the general form x = 5n, y=1-6n is less obvious.

But (to the OP), it's still pretty straightforward: it's clear that 18(15) + 15(-18) = 0, (which would give x = 15n, y = 1-18n). For full generality we divide by the highest common factor of (18, 15), which is 3.
Ah, thanks!
5. (Original post by DFranklin)
I think the question has been badly paraphrased (in particular, what has 14 got to do with the problem as stated...?)

But I think what the OP actually wants to know is (with me giving concrete examples):

Suppose we have a simple Diophantine equation such as 18x+15y = 15.

It's obvious that x = 0, y = 1 is a solution, but the general form x = 5n, y=1-6n is less obvious.

But (to the OP), it's still pretty straightforward: it's clear that 18(15) + 15(-18) = 0, (which would give x = 15n, y = 1-18n). For full generality we divide by the highest common factor of (18, 15), which is 3.
Yeah I guess the question was very badly worded. The given equation was 826x+350y=14 but I wanted to make it more general.

So given the solution pair (-11,26), we can say that 826(-11+25n) +350(26-59n) = 14, for n an integer?
6. (Original post by MR1999)
Yeah I guess the question was very badly worded. The given equation was 826x+350y=14 but I wanted to make it more general.

So given the solution pair (-11,26), we can say that 826(-11+25n) +350(26-59n) = 14, for n an integer?
Looks right.
7. I'm not sure but this may be of help for you:

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