In a linear Diophantine Equation in the form of ax+by=n. Is it possible to find all values of n th

Summer Bradford

Summer Bradford

Answered question

2022-06-07

In a linear Diophantine Equation in the form of ax+by=n.
Is it possible to find all values of n that don't have integer values for x and/or y.
For example 7x+8y=6, x and/or y don't have integer solutions
Are the values of n that don't have integer solutions infinite?
Please this is not a homework
Am just curious.

Answer & Explanation

feaguelaBapzo

feaguelaBapzo

Beginner2022-06-08Added 9 answers

a x + b y = n is a line, thus with the density of continuum.
If you take out the double integral (diophantine) solutions, which may be none, or countable (finite or infinite, depending on the bounds), then you are left, at the minimum, with R Z (eventually, within the given bounds).
pachaquis3s

pachaquis3s

Beginner2022-06-09Added 4 answers

If a and b have a common factor the both ax and by have that factor for all x and y so ax+ by has that factor. If n does not have that factor, the Diophantine equation ax+ by= n has no (integer) solutions. For example 2x+ 6y= 5 has no (integer) solutions.

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