1
$\begingroup$

Problem: The only consecutive non-negative integers a,b and c that satisfy $a^2+b^2=c^2$ are $3,4$ and $5$.

Firstly, How can i turn this into logical statement?
Let $a,b$ and $c$ are consecutive

is it

  1. $a \in \mathbb N, a^2+b^2=c^2 \implies a = 3 $

or

  1. $a \in \mathbb N, a = 3 \implies a^2+b^2=c^2$

I think first one is true, but i have a little confusion..

Then how can i prove it by contradiction?
My attempt:
we know $[(P'\implies q ) \land q' ] \implies P$
we want to show that $a^2+b^2 \ne c^2 \implies a=3 $ is wrong

we get
$a^2 - 2a - 3 = (a-3)\cdot(a+1) \ne 0 \implies a\ne 3$
This completes proof, right?

$\endgroup$
  • $\begingroup$ @MauroALLEGRANZA why did you add $\forall x, y, z$? There is only one $(x,y,z)$ tuple which is $(3,4,5)$ $\endgroup$ – Salih Oct 28 '16 at 12:28
  • $\begingroup$ We may use only $a$, provided that we add the constraint : $b=a+1 \land c=b+1$. $\endgroup$ – Mauro ALLEGRANZA Oct 28 '16 at 12:28
  • $\begingroup$ Because you have to prove that the only solution (with $b=a+1, c=b+1$) is $a=3$; thus : $\forall a \ [a^2+(a+1)^2=(a+2)^2 \to a=3]$. $\endgroup$ – Mauro ALLEGRANZA Oct 28 '16 at 12:30
0
$\begingroup$

You want $a,b,c$ to be consecutive, so

$$a+2=b+1=c.$$

Then your problem can be written

$$a^2+b^2=c^2\text{ and } a+2=b+1=c\implies (a,b,c)=(3,4,5).$$

Edit To answer your other question.

This completes the proof because you are look for non negative integers.

Though there is no need for contradiction here, once you have $(x-3)(x+1)=0$ then you get $a=3$ or $a=-1$, but the case $a=-1$ can be excluded because you are looking for non negative integers.

$\endgroup$
  • $\begingroup$ @E.Joseph thanks for your answer. This completes first part of the question. I will write my attempt for second part now. $\endgroup$ – Salih Oct 28 '16 at 12:44
  • $\begingroup$ In general, is there anything wrong with my approach? Or maybe using '$\implies$' sign etc? $\endgroup$ – Salih Oct 28 '16 at 12:57
  • $\begingroup$ @Salih Nothing is wrong, but you should avoid contradiction when you can to gain in clarity. And about the sign $\implies$, you should only use it when you are writing formally, and avoid it when you are using english words. $\endgroup$ – E. Joseph Oct 28 '16 at 13:00

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.