August eighth is the 220th day of the year, and October 11th is the 284th. If you can’t figure out how those are related, add up all the proper factors of 220; 2, 4, 5, 10, 11, 20, 22, 44, 55 and 110. Now do the same with the factors of 284; 1, 2, 4, 71, and 142. Numbers with this property are called amicable numbers, and they date back to antiquity. The relationship between 220 and 284 was known at least as far back as Pythagoras (500 BC). Sam Kutler has written to tell me that the first use of a term like "friend" for the pair was in a commentary on the work of Nicomachus by Iamblichus, around 300 AD. He also thought the Greek term was arithmoi philos, literally, friendly numbers. 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. It is likely that the ancients assumed that this pair was unique and there were no others. Western mathematics knew only this set until 1636 when Fermat discovered a second pair; 17,296 and 18,416 (I leave the proof that they are amicable as an exercise for the reader, 
). After his discovery, the search for amicable numbers became quite a trend among mathematicians of the time. Descartes discovered a third pair and Euler added over sixty more pairs to the known list. The second smallest pair,1,184 and 1,210, was overlooked by all of these people, and found by a sixteen year old Italian student, Nicolo Paganini, in 1866.
Arabic mathematicians had preceeded their western counterparts by many hundreds of years. Thabit ibn Qurra discovered, and proved, a rule for creating amicable pairs in the ninth century. The St Andrews web page tells us that :
Al-Farisi (born 1260) gave a new proof of Thabit ibn Qurra's theorem, introducing important new ideas concerning factorisation and combinatorial methods. He also gave the pair of amicable numbers 17296, 18416 which have been attributed to Euler(I have only seen this creidited to Fermat, as I have above. Not sure if they have made a mistake (shudder) or know something I don't (more likely))., but we know that these were known earlier than al-Farisi, perhaps even by Thabit ibn Qurra himself.
You can find a rather extensive list of known amicable pairs here.
There are chains of numbers in which the sum of the principal factors of one number sum to the next. Groups of such numbers are called “sociable” numbers. One chain of sociable numbers is given by 12496, 14288, 15472, 14536, and 14264.
Amicable comes from the Latin word amicus for friend and the related word amare for to love. Some other words from the same root include amigo, amateur, and amour.
August eight is also the day Tycho Brahe laid the cornerstone for the observatory he called Uraniborg on the island of Hven. Tycho was not only a great astronomer, perhaps the last great naked-eye astronomer, but he also was missing a nose. Ok, actually he had several, one of which was made of gold, but he didn’t have one of flesh like we all do (at least I hope you still have yours). It seems he was in a duel as a college student and lost his flesh one, so he adopted a number of prosthetic replacements that he wore for the rest of his life. There is another great mathematician who spent much of his life without his nose. You can find a clue here.
Today is also the date on which Danilo Blanusa died in 1987. If you never heard of this Croatian Mathematician or his work, you are not alone. I remember him only for the interesting name of the mathematical graphs he worked with, snarks. A snark is a special kind of non-planer graph (you can’t draw it on a piece of paper without having edges cross, in fact the simplest snark, picture above,requires at least two crossings. A snark is a graph that is connected (you can get from any point, or vertex, to any other), every node has three branches, and it has a chromatic index of four ( The rules for a snark played a part in the proof of the famous four-color theorem. The first known Snark shown above, is called a Petersen Graph, and was discovered in 1898. P.G.Tait had shown that the four color theorem was equivalent to the question of whether a Snark was Plainer, and Petersen was the first to find an example. Blanusa found two more (1946), and a generalization of his method proved that there were an infinite number of them (1975).
The name Snark was created by Martin Garnder after the elusive object of the hunt in Lewis Carroll’s “The Hunting of the Snark”.
The name Snark was created by Martin Garnder after the elusive object of the hunt in Lewis Carroll’s “The Hunting of the Snark”.
220, because it is divisible by the sum of its digits, is also a Harshad number. Harshad numbers were defined by D. R. Kaprekar, (and are alternately called Niven numbers). Kaprekar is an Indian mathematician, and created the term “Harshad” from the Sanskrit root har sa, which means “great joy.” Kaprekar is remembered for Kaprekar numbers and is also remembered for discovering that if you take a four digit number, write the digits in ascending and descending order and subtract, and then repeat the process with the new result, eventually you will end up with 6174. A similar method works for numbers with two or more digits. This process is called the Karprekar process. For example if we start with 3245 we write the digits in order both ways 5432 - 2345 and get 3087. Now we repeat the process with 3087 to get 8730-0378 = 8352. Continuing the process we get 6174 for the next result. It is an interesting exercise for elementary students to try to find the length of the Kaprekar sequence for three and four digit numbers (and they never realize they are practicing subtraction skills).
220 is also interesting as it is the sum of four consecutive primes (47 + 53 + 59 + 61) and is also the sum of the first ten triangular numbers (1+3+6+10+15+21+…+ 55)
It will be exactly 11 days before the date which is the sum of the first eleven triangular numbers August 17th and 20 days before the next day that is the sum of four consecutive primes, August 28th ..
The sequence 220 first occurs in the decimal expansion of Pi starting at the 1910th decimal place.
Eight is a lucky number in China, and so the government is auctioning off license plates with eights and other lucky numbers. You can read a blog from the Freakonomics web site on this Chinese fascination with eight. They point out that, “… the Chinese consider 8 to be such a lucky number that the Beijing Olympics are due to open on 8/8/08 at 8 p.m.”
