In Godel, Escher, Bach: An Eternal Golden Braid, Douglas Hofstadter outlines his axiomatic “Typographical Number Theory” and sets certain problems for the reader – one is to show that **b** (a variable) is a power of 2.

These are my notes in trying to demonstrate this (hopefully understandable to anyone who knows a bit of symbolic logic and has a passing knowledge of Peano’s axioms.

My reasoning runs like this: if a number is a power of two then it has no prime factors other than 2 (and 1, if we count that as a prime) – see the earlier entry on the fundamental theorem of arithmetic.

A prime (a) in TNT:

**First Update**: Now realise there is a simpler way of stating a prime in TNT – namely that there is no two numbers bigger than one the product of which is the number we seek:

**Second update:**

So, we have a definition for a prime, **a**, and we can set **a** to a prime other than 2.

Then

~~I ~~*think* this is a sufficient definition for “b is a power of 2”, as we know that all numbers are products of primes and we have ruled out any prime other 2 being a factor of b. Is that right?

On further thought I don’t think that’s enough, because we need to specify that **b ** is a multiple of 2.

So what about this?

**So has that nailed it?**

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- How I discovered the fundamental theorem of arithmetic by chance (cartesianproduct.wordpress.com)