Prime Edition

ROHIT GUPTA
SPACE MONKEY

Rohit Gupta (@fadesingh on Twitter) is a mathematician who thinks the universe is a hologram projected by a microscopic disco ball.

Rabbits on the moon and the Black Hole of Wall-Sun-Sun

Chinese myth suggests that a Jade Rabbit lives on the Moon, pounding away at lunar regolith to extract the elixir of life. If such an immortal rabbit were to reproduce, and its children multiply without dying, one would see the population of rabbits on the moon evolve like this - 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610 and so on till infinity. The Apollo mission found no such creature on the moon, however...

Two consecutive numbers when added result in the next number in the sequence above, observed first by Pingala (200 B.C.) and later, independently - in the middle ages by Leonardo of Pisa, better known as Fibonacci. 1+1 = 2, 1+2 = 3, 2+3=5, 3+5=8...thus do these breeder bunnies grow. For brevity, let us say that F(n) denotes the n'th number in this sequence and observe the table pictured above.

By themselves, these numbers are not so interesting - unless you can see an underlying pattern. Prime numbers are the atoms which help us define a unique formula for each integer (see: "The Fundamental Theorem of Arithmetic" on Wikipedia). And this is where we find a highly structured landscape, full of periods, connections and loops. Take F(33), the 33rd Fibonacci number and its formula in primes - 2 X 89 X 19801. Before we go further, keep in mind that 33 = 3 X 11.

The number 2 is actually F(3) and 89 is F(11), so we learn here that when m divides n, then F(m) definitely divides F(n). The number 19801 is in green because it appears in a formula in this sequence for the very first time. This is true for all the primes in green. In fact, you will notice that every new Fibonacci number in the sequence brings at least one new prime number with it which has not been seen before. Sometimes, and this is extremely rare, the Fibonacci number itself is that prime. Also, no two consecutive numbers have any factors in common.

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A solution to the Wall-Sun-Sun problem will offer an elementary proof of Fermat’s Last Theorem, which will be much applauded.

ow dwell on the following question for a few minutes - whenever p is a prime number (greater than 5, marked in red) try to scan closely the neighbourhood of F(p). Do you see a pattern jump at you? There it is, again! Either in one number before or the one after F(p), the same prime p appears as a factor. You can find tables for prime factors of thousands of Fibonacci numbers on the internet and check this to be true, always.

In 1960, D.D. Wall asked whether this p will never appear more than once within these factors. For instance F(30) contains the factor 2 thrice ( therefore, raised to the power 3) , but 31 appears only once. Why is this question important? For one thing, it is still an unsolved problem after half a century, and goes by the name of The Wall-Sun-Sun Prime Conjecture. It is suggested by some that there are an infinite number of such primes, but astronomical computations have revealed none so far. It is as if someone has predicted the existence of a black hole in a neighbouring galaxy but none can be detected.

In 1992, the Chinese twin brothers Zhi Hong Sun and Zhi Wei Sun showed a startling connection between these exotic primes and the famous Fermat's Last Theorem. The historic theorem was later proved by Andrew Wiles, but the proof is so long and complicated that only the best mathematicians fully understand it. I'm certainly not one of them, not yet at least. Some authors suggest that a solution to the Wall-Sun-Sun problem will offer a much shorter and perhaps elementary proof of Fermat's Last Theorem, which will be much applauded.

Li Bai, a famous poet of the Tang Dynasty, wailed that, "The rabbit in the moon pounds the medicine in vain." Perhaps that may be true of the rabbit, Mr. Li Bai - but giving up is not the style of space-faring monkeys.

 
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