Well, I really liked this problem and I would like you all to give it a try...

S is an infinite set of points in the plane. The distance between any two points of S is integral. Prove that S is a subset of straight line.

Hint - Use contradiction

[Putnam 1958/B5]

Question

Well, I really liked this problem and I would like you all to give it a try...

S is an infinite set of points in the plane. The distance between any two points of S is integral. Prove that S is a subset of straight line.

Hint - Use contradiction

[Putnam 1958/B5]

anonymous · Answer

yeah... thats over my head lol

mathslover · Answer

@vishweshshrimali5 - What does this mean - "The distance between any two points of S is integral." ? 

Does this mean that we can write the distance as an integral of a function or something else...?

vishweshshrimali5 · Answer

Integral here means that the distance is an integer

mathslover · Answer

Oops. Okay! :)

kainui · Answer

I am trying to figure out a counter example that involves using pythagorean triples since that would be the only way I can think of haha. Hmm interesting problem.

kainui · Answer

Ok so there are an infinite number of primitive Pythagorean triples. We can take one of these such as (3,4,5) and (5,12, 13) and draw two triangles like this |dw:1427088499393:dw| Now notice we can multiply the vertical leg on either one by the other's number so we have 3*5 and 5*3 and we can draw this: