Some Proofs are Prettier Than Others

Just read a post about Nichomachus's Theorem at as a guest blog on Loren Shure's Matlab blog. It pointed out that the theorem was cleverly proven by the simple picture below.
The idea of a proof without words is that you can "see" what it shows.

The theorem is usually presented in pre-calculus classes when they cover sequence and series. In simple words it says the square of the nth triangular number is equal to the sum of the cubes of the first n integers. Cool math language to replace all that verbiage is .

Nicomachus lived in what would now be part of Jordan around 100 AD, and was a follower of the Pythagorean Cult. He wrote about arithmetic and his works were translated into Latin by Boetheus. His book on music and its relation to math is the earliest source of the story that Pythagoras came up with the idea of harmonic tones when he walked past a blacksmith pounding on an anvil. His "Art of Arithmetic" contains the equality above.

I like the "proof without words", but I think my favorite was by Charles Wheatstone. I first heard of Wheatstone as a young electronic trainee in the Air Force. He didn't invent, but did improve a device for measuring the resistance of ..well, almost anything, which is now called the Wheatstone Bridge. He also invented the Playfair cipher (math names are confusing, but Lord Playfair was a heavy promoter of the cipher), the concertina and the stereoscope (if you are older, you looked at cards at your grandmothers house through one of these that gave a 3-D view of the Taj Mahal or other beautiful scenery. He also came up with the following really clever proof of Nicomachus's theorem.

1 + 8 + 27 + 64 + 125 + ...
= (1) + (3 + 5) + (7 + 9 + 11) + (13 + 15 + 17 + 19) + (21 + 23 + 25 + 27 + 29) + ...
= 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 + 21 + 23 + 25 + 27 + 29 ...

and since the sum of the first n odd integers is equal to n^2, we only have to see that by the method he has broken up the cubes there will be 1+2+3+...+n odd integers for the first n cubes... which seems really nice to me.

The NEW Friendly Numbers

Friendly Numbers

Up until the spring of 2008, if you asked me what "friendly" numbers were, I would refer you to my listing for amicable or aliquot numbers. Two numbers were amicable (friendly) if the sum of the factors of each equaled the other. 220 and 284 are the oldest pair known, and they date back to Pythagoras. The numbers were inscribed on "magic charms" in the middle ages which were sold to insure the fidelity of ones lover. Other stories suggest that the gift of 220 Goats from Jacob to Esau in the Biblical story was an expression of love made significant by the use of one of the pair. The proper divisors of 220, 1, 2, 4, 5, 10, 11, 20, 22, 44,55, and 110, add up to 284 and the same is true the other way around. Until very recently, at least to the best of my knowledge, this was what people meant when they referred to "friendly" numbers. This is how Simon Singh describes them in his book Fermat's Enigma; Western mathematics knew only the one pair until 1636 when Fermat discovered a second pair; 17,296 and 18,416 . This was also how Hoffman described them in his book about Paul Erdos, The Man Who Loved Only Numbers, and how Alfed Posamentier had used the term in Math Charmers: Tantalizing Tidbits of the Mind. These were world class mathematician/authors. I assumed I was in line with the current usage.

Then recently, I came across a reference on Mathworld that described them as the ratio of the sum of all the divisors (including the number itself) divided by the number. For example, 8 can be divided by 1, 2, 4, and itself, 8. The sum of its divisors is 15, so its ratio, is 15/8; this is sometimes called the "abundance" of the number. Do not confuse this with the much older term "abundant" for a number for which the sum of the proper divisors (factors, or divisors not including the number itself) is greater than the number itself.. For example the proper divisors of 8 are 1, 2, and 4, which total 7, so 8 is NOT abundant, but deficient.

The classification of numbers as being deficient (the sum of the proper divisors is LESS than the number), abundant (the sum is greater than the number) and perfect (the sum is equal to the number, as in 6 and 28) goes back at least to Nicomachus (about 100 CE) who separated the even numbers into abundant (it was the mistaken belief for a long time that all odd numbers were deficient) or perfect.

When the ratio of the sum of the numbers divided by the number equals two, the number is perfect, so all perfect numbers are friendly under this new usage. Other numbers that are mutually friendly besides 6, 28 and the rest of the perfect numbers include 30 and 140, with a ratio of 12/5, as well as 80 and 200, whose ratio should be 2.325, if I calculated right. There are many numbers, such as all the primes, that are known to be solitary, that is, they have no friends. All the primes have a ratios of (p+1)/p, so their ratio would get smaller towards a limit of 1 as the size of the prime grew larger. There are other numbers, some relatively small like 10 and 14, that we do not know if they are friendly or solitary. I wondered as I computed these if there is a number or a pair with the greatest ratio? (write if you know, please)

I am not sure, and am presently searching to find the first use of the newer use of friendly numbers, but it seems to exist at least since the 1970's (surprised me!) from a citation on Sloans integer sequence site for ...Anderson, C. W. and Hickerson, D. Problem 6020. "Friendly Integers." Amer. Math. Monthly 84, 65-66, 1977. Ok, if you know about this stuff, drop me a line and set me straight.