perplexus.info :: Just Math : Some Lengths May Parallel X-axis

Some Lengths May Parallel X-axis (Posted on 2023-07-09)

Let F(x) be a polynomial with integer coefficients.

There are two distinct points on the graph of F, say P and Q, with integer coordinates.

If the length of PQ is an integer, then will PQ always be parallel to the x-axis?

If so, prove it.
If not, provide an example.

See The Solution Submitted by K Sengupta

Rating: 5.0000 (1 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)

re: Seems unlikely

| Comment 2 of 3 |

(In reply to Seems unlikely by broll)

4x-3y=0 is not written as F(x). If y=F(x) then the coefficient of x is 3/4 which is not an integer.

The linear case is the easiest to prove: a non-zero integer slope requires a PPT with a side length 1.

I suspect the answer is yes.

Posted by Jer on 2023-07-10 08:58:37

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