Answer to the Pythagorean Problem

The answer to a problem incliuded in yesterday’s post Monday Miscellany.

INTRODUCTION

You may remember that in yesterday’s post titled “Monday Miscellany” I included one of the problems I had solvced on the mathematical website Brilliant – this post presents the solution and also clears up a side issue raised on that website by disgruntled folk who had got it wrong.

THE PROBLEM

Here is the problem again from yesterday:

Pythag

THE SOLUTION

The answer is True. The formula for (x+1)^2 is X^2 + 2X + 1, and every odd number greater than 1 could serve as the 2X+1 part of that equation. 

THE SIDE ISSUE

Some people on Brilliant cavilled at this because there are some Primitive Pythagorean Triples whose smallest term is even (8,15, 17, 20,21,29 and 65,72,97 were all mentioned, although none of the complainers mentioned 12,35,37, 60,91,109 or 696,697,985). The question did not state that the triple of which the odd number is the lowest term was the lowest triple to feature that number, and indeed if one looks carefully at the triangles presented as part of the problem one can see clearly that the odd number is allowed to be in another triple where it is not the lowest term:

Pythagorean Triples Solution - a

Note that the number 5 features twice (ringed in the diagram above, once as the largest term in a triple and once as the smallest).

Thus that 15 features in 8,15,17 does not invalidate the claim of the question since it is the smallest term in 15, 112, 113. All the other odd numbers mentioned in triples of which they are not the smallest member likewise feature in triples in which they are the smallest member, the biggest being 985, 485112, 485113. 

Author: Thomas

I am branch secretary of NAS West Norfolk and #actuallyautistic (diagnosed 10 years ago at the comparatively advanced age of 31). I am a keen photographer, so that most of my own posts contain photos. I am a keen cricket fan and often write about that subject. I also focus a lot on politics and on nature.

3 thoughts on “Answer to the Pythagorean Problem”

  1. At first the question looked a bit tough and strange.. but once the soultion got visible , I’m amused at myself & yeah it was logical and simple . Thnx for this interesting post.. quite rare on WP !

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