# [FOM] Historical queries

joeshipman@aol.com joeshipman at aol.com
Wed Nov 14 10:26:33 EST 2007

```I wrote:

>-----Original Message-----
>From: joeshipman at aol.com
>In a forthcoming paper cowritten with John H. Conway, I show that, for
>any non-square integer N, there is a very simple geometric proof that
>the square root of N is irrational that could have been demonstrated
by
>Theodorus; however, to find the proof requires finding a number of the
>form (k^2)+1 or (k^2)-1 that is divisible by N, and the guarantee that
>such a number exists depends on the theory of the Fermat-Pell
equation,
>which no surviving manuscripts indicate was known by the ancients
>(although Diophantus and Archimedes may well have known this fact
>anyway).

I should clarify that the proof requires (k^2)+1 or (k^2)-1 to be a
square times N, not just divisible by N. It shows that the square root
of any integer k^2+1 or k^2-1 is irrational, from which one obtains the
irrationality of any square root of an integer which can be multiplied
by a square to get within 1 of a square.

Here is the solution to the historical mystery of why Theodorus stopped
at 17:

n Size of triangle (relative to base) needed for proof
2  Side 1
3  Hypotenuse 2
5 Side 2
6 Hypotenuse 5
7 Hypotenuse 8
8 follows from 2
10 Side 3
11 Hypotenuse 10
12 follows from 3
13 Side 18
14 Hypotenuse 15
15 Hypotenuse 4
17 Side 4
18 follows from 2
19 Hypotenuse 170

-- JS

________________________________________________________________________
Email and AIM finally together. You've gotta check out free AOL Mail! -
http://mail.aol.com
```