large_primes
2.X.X
Large Prime and Perfect Numbers
Introduction
In antiquity science, mathematics, and mysticism were inextricably linked. As such astrology lived alongside astronomy, and numerology and what we now call number theory lived alongside arithmetic, algebra, and geometry. The greeks, who laid the foundations of axiomatic geometry, were also mystified by all sorts of patterns and relations they saw in numbers. They began the study of prime numbers, perfect numbers, amicable numbers, triangular numbers, and many more. While numerology, the practice of associating mystical meanings to numbers and using them to foretell the future, has gone the way of astrology, many of the purely number theoretic properties the greeks identified, which weren't strictly mystical, have remained to this day in a branch of mathematics known as number theory. While number theory has some important results related to encryption, and is also a fascinating subject in it's own right, our primary interest here is that number theory often leads to some tremendous large numbers quite naturally. That's because number theory is mostly concerned with the integers, and the positive integers in particular. We can say that number theory is the study of the positive integers, and the properties that some of it's members possess. For any given property, we can ask a few basic questions (1) does any positive integer possess it (2) if such a number exists what is the smallest example of such a number (3) if more than one positive integer possesses a property than how many do, and is this number finite or infinite (4) what is the first member of this sequence, 2nd, 3rd, 4th etc. (5) What's the largest example of the sequence we know of or could give? Note that in the case of the last question the answer must be a finite number, because all positive integers, no matter how large, are finite. A natural first step to analyzing any particular property is to either (1) find at least one example of a number with such a property or (2) to simply begin testing the positive integers in sequence starting with 1 and working your way up. Since there are only a finite number of positive integers less than any given integer, c, we eventually exhaust all the integers less than c, and curiousity therefore naturally leads us towards positive infinity. Hence, there is much in number theory that is potentially appealing to the googologist.
The amount of number theoretic properties that have been studied is overwhelming! We are only going to focus on the most prevalent properties here, with an eye towards giving large number examples. Part of the game here is not simply to start off with a trivial property and then come up with a large example. For example we can generate arbitrarily large even and odd numbers, but this isn't "interesting". What we want to look into is properties with a certain degree of rarity and difficultly, and then demonstrate very large examples of those properties. Believe it or not this task can become non-trivial, which means it makes sense to speak of the "largest known number" with some specific property.
The two main ones we'll be looking into is large prime numbers, and large perfect numbers. Before we jump into that however, I'll provide a little background into what these kinds of numbers are and what makes them of some "interest".
Primes and Perfects
Although ancient people were probably aware of the existence of prime numbers, the earliest records we have of a serious study of them begins with the greeks [1]. Euclid proved that there are an infinite number of prime numbers, and also proved the fundamental theorem of arithmetic which states that all positive integers greater than 1 are either prime, or unique products of primes. Not only do these two mathematical facts form the basis of all number theory, but they also provide the foundation for computer encryption. Another reason why primes are of particular interest is because they exhibit apparently random behavior, even though they exist in the abstract and deterministic world of mathematics. This is part of their allure and mystery. The greeks were no doubt bewildered and fascinated by this strange property, but before we get ahead of ourselves, what are prime numbers? A prime number is a positive number greater than 1 whose only factors are 1 and itself.