Last week we looked at some of the characteristics of the versatile number 8, which is, among other things, a leyland number. Let’s see what our regular commentator Bretos Bursó says about it:
“The sequence of Leyland numbers appears in the OEIS, but it adds the number 3 as the initial element, without giving an explanation. Answering Carlo’s question is easy: delete x=1 or y=1 because otherwise every number n would be a Leyland number (n=1^(n-1)+(n-1)^1). In the OEIS entry there is a link to a text file with the first 5,000 Leyland numbers, and it has come to my attention that the number 98 is 20000000000. I ask the obvious question: knowing that a Leyland number is equal to x^ y + y^x, with x and y integers greater than 1, how do you find what x and y are? For some it is very easy, but how would it be solved, for example, for the one in place 100, which is 31381070257?
The OEIS is the Online Encyclopedia of Whole Number Sequencesand I was also surprised that the 3 at the beginning of the list, since in all the articles that I had read so far about the Leyland numbers, 8 is considered to be the first of them. Can you think of an explanation for this inclusion? And, given a Leyland number, how can we find the x and y that generate it?
Within the Leyland numbers, those that are prime are of special interest, especially for their use in cryptography. The smallest Leyland prime is 17, the second 593 and we don’t find another one until we reach 32993, from where we jump to 2097593: Leyland primes are widely spaced and their sequence grows very quickly. The largest known Leyland prime is the one corresponding to the values 2929 and 8656 for x and y, a number of 30008 digits.
There are also, among the Leyland numbers, large probable primes, such as 9 and 314738. As the name suggests, a probable prime is a number that is very likely to be prime, even though it has not been proven to be so. The probable primes pass Fermat’s primality test, based on his “little theorem”.
A not-so-small theorem
Fermat’s Little Theorem says that if to is a positive integer and p a cousin that is not a factor of toso p must be a factor of aᴾ⁻¹ – a. For example, yes to = 8 and p = 3, we see that 8² – 1 = 63, and 63 is divisible by 3. How can we base a primality test capable of detecting probable primes on this theorem?
It is called “Fermat’s little theorem” to distinguish it from what is known as Fermat’s last theorem and, today, as the Fermat-Wiles theorem, since it was proved in 1995 by the British mathematician Andrew Wiles. Said theorem affirms that it is not possible to find three positive integers x, y, z such that they verify the equation, xⁿ + yⁿ = zⁿ, for no greater than or equal to 3. What seems like an innocent extension of the Pythagorean theorem turns out to be impossible; an impossibility so difficult to prove that mathematicians have taken more than three centuries to achieve it. Wiles himself described his process thus:
“You enter the first room of a mansion and it is dark. You keep bumping into the furniture, but little by little you learn where each piece of furniture is. Finally, after six months or so, you find the light switch and suddenly everything lights up. You can see exactly where you are. Then you go to the next room and spend another six months in the dark. Thus all these advances, though sometimes very rapid and accomplished in a day or two, are the culmination of preceding months of stumbling in the dark, without which the advance would have been impossible.”
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