## Sunday, October 26, 2014

### Some interesting properties of numbers

In  Fermat's Last Theorem, Simon Singh writes about some interesting patterns among numbers:

1. Perfect numbers - Numbers whose divisors add up to the number itself. Eg.6 has divisors 1, 2 and 3 which add up to 6. The next perfect number is 28. If the sum of the divisors is more than the number, Pythagoras called it an 'excessive' number and if the divisors added up to less than the number, he called it 'defecttive'.
2. Friendly numbers or amicable numbers are closely related to perfect numbers.They are pairs of numbers such that each number is the sum of the divisors of the other number. For eg. 220 and 284 are friendly numbers. 220 is the sum of the  divisors of 284 (1, 2 4 71 and 142). 284 is the sum of the divisors of 220 (1, 2, 4, 5,10, 11, 20, 22, 44, 55and 110). Fermat discovered 17,296 and 18,416. Descartes discovered a third pair (9,363,584 and 9,437,056). Leonhard Euler discovered 62 pairs. Strangely,they all had missed a much smaller pair which was discovered by a 16 year old Italian - 1184 and 1210.
3. Sociable numbers are 3 or more numbers which  form a closed loop. Consider the loop of five numbers: 12,496; 14,288; 15,472; 14,536; 14,264. The divisors of the first number add up to the second, the divisors of the second add up to the third, the divisors of the third add up to the fourth, the divisors of the fourth add up to the fifth, and the divisors of the fifth add up to the first.
4. Fermat proved that 26 is the only number sandwiched between a square and a cube (between 25=52  and 27= 33)
5. All prime numbers (except 2) can be placed in two categories: those  which can be written as 4n + 1 and those which can be written as 4n - 1 where n equals some number.Thus 13 is in the former group (4*3 + 1) while 19 is in the latter  group (4*5 - 1). Fermat's prime theorem claimed that the first type of primes were always the sum of two squares while the second type could never be written in this way. The theorem was proved by Euler almost a century after Fermat's death.

PS: Since I will rarely do any maths related posts (I know that you are dreadfully disappointed but as a wise philosopher of yore said, you can't always get what you want) I am generally giving a couple of links about maths that I had saved:

1. An excellant series in NYT by Steven Strogatz on The Elements of Math
2. A talk by Simon Singh about "The Simpsons and Their Mathematical Secrets