Tag Archives: Euclid

Test Your Intuition (16): Euclid’s Number Theory Theorems

Euclid’s

Euclid’s book IX on number theory contains 36 propositions.

The 36th proposition is:

Proposition 36.If as many numbers as we please beginning from a unit are set out continuously in double proportion until the sum of all becomes prime, and if the sum multiplied into the last makes some number, then the product is perfect.

It asserts that if 2^n-1 is a prime number then 2^{n-1}\cdot (2^n-1) is a perfect number. (A number m is perfect of it is equal to the sum of its proper divisors.)

This is certainly a remarkable achievement of ancient Greek mathematics. Other Propositions of the same book would be less impressive for us:

Proposition 23.If as many odd numbers as we please are added together, and their multitude is odd, then the sum is also odd.

Proposition 24.If an even number is subtracted from an even number, then the remainder is even.

Proposition 25.If an odd number is subtracted from an even number, then the remainder is odd.

Proposition 26.If an odd number is subtracted from an odd number, then the remainder is even.

Proposition 27.If an even number is subtracted from an odd number, then the remainder is odd.

Proposition 28.If an odd number is multiplied by an even number, then the product is even.

Proposition 29.If an odd number is multiplied by an odd number, then the product is odd.

Test your intuition: What is the reason that deep mathematical results are stated by Euclid along with trivial results.