High Numbers.....Who?

What is the highest number?

Stupid question, really, as there is no "highest" number.

Alright then, let us qualify it a bit.

What is the highest number that has ever been used, and can easily be understood? (I put in that last caveat "easily be understood", as Graham's number is much, much larger, but requires far more algebra to understand)

To answer that, a little background is necessary. (For you guys out there who know I am at least modestly adept in algebra, you can skip this, but I know you will read it anyway. Hey! I have said this before!)

The background needed is about the prime numbers. Prime numbers are numbers that cannot be divided by anything exactly, other than themselves and 1. For example, 12 is not a prime number because it is divisible by 1,2,3,4,6 and 12. However, 13 is a prime number, because it can only be divided by 1 and 13.

So far so good.

Many math guys have tried to find formulas (formulae, atch) to generate prime numbers, the two simplest being "4n+1" and "4n-1" where "n" is an integer (whole number). These two yield prime numbers sometimes, but not always, and the race is ON! Which will bring in the most prime numbers?

Well, up to n = 125,000, there is a clear winner, but after that, the winner oscillates back and forth, much like a rather close horse race.

Background over. Serious stuff now. It has been proved that the "lead" will always oscillate until a certain number has been reached, then one will dominate and probably win. This is the number I am talking about.

The answer to the question:

What is the highest number that has ever been used, and can easily be understood?

It is up to now, this number. Let's call it "F". It is not less than 10 to the power 10 to the power 10 to the power 34.

Fairly big number.

Allow me to illustrate how big it actually is.

To conceptualise high numbers (who?) there needs to be some sort of measuring stick. The measuring stick I choose is the
"average number of distinct prime factors". Now, what is a prime factor?

A prime factor is a prime number which exactly divides the number in question. For instance, 6 has factors which divide it exactly to be 1, 2, 3 and 6. Since 2 is by definition a prime number, and we don't count 1, the number of prime factors of 6 is two (6 is not a prime number). How about 28? Well, the numbers that divide 28 exactly are 28, 14, 7, 2, and 1. Only 2 and 7 are prime numbers. That makes two prime factors.

How about 37, a prime number?

Only 1 and 37 exactly divide this prime number, so the number of distinct prime factors is one only (37).

So, in the run up from 1 to 100, what is the "average" number of distinct prime factors for this range? Turns out, it is only about 1.9.

How about the range 1 - 1000? The "average" number of distinct prime factors, (lets call it AP(n)) jumps to about 1.95. The occurrence of the prime numbers brings this average down.

How about 1 - 1,000,000? AP(n) climbs to about 2.3.

Let's get really bold = 1 - googol (a googol is 1 followed by 100 zeroes). Yes, a significant jump.....to AP(n) = 19.

Now here is the measure that I was talking about. What if we average out the number of distinct prime factors for the numbers between 1 and F (10 to the 10 to the 10 to the 34, the number earlier outlined).

What is AP(n) for n = F? It is truly a measure of the magnificence of F when we discover that AP(F) is a number in excess of 10 to the 24! 1 followed by 24 zeroes! That is, there are , on average, 1,000,000,000,000,000,000,000,000 distinct prime factors in the numbers preceding F.

Remember, this is only the "average"!

Isn't math wonderful, especially with The High Numbers (Who?)?