A Criteria of Diophantine Equations on one variable, Is it wrong?

Question

The following is a Home Work question provided by our instructor:

let a,b,c be positive integers. Prove that if a+b>c then the equation ax+by = c has no solutions in positive integers.

I think the statement is false since $4x+12y=4$ is a counter example. However, it is wired that a home work question be formulated that way. Or, did I miss something?

Anurag already gave a hint (except it should've had $\geq$, not $>$) — Wojowu, Oct 03 '18 at 20:32
@AnuragA I thing you are clarifying something. I will try to have a second deep look. Thanks,, — Maged Saeed, Oct 03 '18 at 20:33
@BillDubuque: I've heard of zero being considered as a natural number at times, but never positive. What language(s)? Do you have a reference? — RghtHndSd, Oct 03 '18 at 21:20

score 1 · Accepted Answer · answered Oct 03 '18 at 20:33

1

The instructor is interested in positive integers,which are {1,2,3,4,....}

If $ a+b>c$ and $x\ge 1$ and $y\ge 1$, then $ax+by \ge a+b >c$

Thus the equality does not happen, that is there is no solution in positive integers.

answered Oct 03 '18 at 20:33

Mohammad Riazi-Kermani

68,728

Thanks for clarifying. – Maged Saeed Oct 03 '18 at 20:37

A Criteria of Diophantine Equations on one variable, Is it wrong?

1 Answers1