Number theory is an interesting branch of mathematics that deals with the study and analysis of integers and is primarily done for research or academic purposes. Recreational number theory on the other hand is mostly done by amateurs such as myself for fun and can involve finding patters and solving puzzles using numbers. One can slot number theory into three broad categories. The first being classic or elementary number theory, The second is algebraic number theory and lastly analytic number theory. The List of recreational number theory topics we will discuss herein purely deals with classic number theory.The dutch mathematician Hendrik Lenstra with tounge firmly in cheek stated that "Nowadays, when a Number Theorist applies for a grant, he says that Number Theory is used in cryptography, and so doing Number Theory is good for National Security. Back then, since it was before the discovery of America, they said Number Theory is used in music. But I won't comment on the progress of civilization since then:"
Perfect Numbers
They were most probably first discovered in the time period that spanned the 7th century BC and the 4th century AD by the then Greek speaking world. A perfect number can only be a positive number that equals the sum of its divisors, excluding itself. In the first examples listed the integer 6 is not only divisible by 1, 2 and 3 but the sum of 1,2 & 3 is also equal to 6.
6 = 1+2+3
28 = 1+2+4+7+14
496 = 1+2+4+8+16+31+62+124+248
Narcissistic Numbers
Greek mythology states that Narcissus who was the son of river god Cephissus and nymph Liriope was a person of exceptional beauty. Narcissus ended up dying because he was unable to leave a pool after falling in love with his own reflection. we also happen to have numbers that can’t get enough of themselves. They happen to be numbers that are the sum of its own digits when each digit is raised to the power of the number of digits. By that definition all one digit numbers are narcissistic. There are no two digit narcissistic numbers but there are four three-digit narcissistic numbers.
370 = 3^3 + 7^3 + 0^3
1^3 + 5^3 + 3^3
Mersenne Primes
Marin Mersenne who "Mersenne Primes" were named after is a very interesting individual. Mersenne was an ordained priest and a friar who while in Paris at the convent of the annunciation met other greats like René Descartes and Étienne Pascal (father of Blaise Pascal) who influenced his work. Mersenne also seemed to be a man of strong convictions who was an ardent supporter of Galileo during the geocentric vs heliocentric debates even though he was a Catholic. Mersenne primes are prime numbers that fit the pattern 2^p-1, where p is any prime number. An example is where p=2, therefore 2^2-1 = 3. Three happens to be a prime hence it is a Mersenne prime. Other Mersenne primes include
p = 3, therefore 2^3-1 = 7
p = 5, therefore 2^5-1 = 31
p = 7, therefore 2^7-1 = 127
p = 11, therefore 2^11-1 = 2047
Dudeney Primes
Henry Dudeney was an English civil servant who at a very young age was much inspired by his grandfather who was also a mathematician. He was considered one of Englands premiere recreational number theorists and puzzle maker. Dudeney noticed that some numbers exhibit a behavior where the numbers individual digits sum add up to their cube root. There are only six known Dudeney numbers until someone proves this wrong.
1) number 1 where, 1 = 1, and 1*1*1 = 1
2) number 512 where, 5+1+2 = 8, and 8*8*8 = 512
3) number 4913 where 4+9+1+3 = 17 and 17*17*17 = 4913
4) number 5832 where 5+8+3+2 = 18 and 18*18*18 = 5832
5) number 17576 where 1+7+5+7+6 = 26 and 26*26*26 = 17576
6) number 19683 where 1+9+6+8+3 = 27 and 27*27*27 = 19683
Twin, Cousin & Sexy Primes
I first heard about twin, cousin and sexy primes while watching an episode of the "Colbert Report" when they had on Terence Tao the mathematical genius from Caltech. Twin primes are when the difference between two consecutive primes is equal to two. Examples of twin primes are:
(3,5) (5,7) (11,13) (17,19) (29,31)
Cousin primes are when the difference between two consecutive primes is equal to four. Examples of cousin primes are:
(3, 7), (7, 11), (13, 17), (19, 23), (37, 41)
Finally Sexy primes are when the difference between two primes is equal to (you guessed it) six:
(5,11), (7,13), (11,17), (13,19), (17,23)
The recreational number theory topics listed above are the ones that intrigued me and more importantly the ones that I could understand. If this is something you like, you can lookup a fairly exhaustive list of topics listed on wiki which can be found here (https://goo.gl/OVVH0m).
