The Prime

One of the most interesting types of numbers is the prime. A prime is a number that is only divisible by itself and one. 1 is the identity and is not considered a prime. The primes under 100 are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, and 97. There are 25 primes under 100, but between 100 and 200, there are 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, only 21 primes. Primes go in no definite pattern, but there is an easy primality test that can be performed by calculator. The is called the sieve of Eratosthenes.

For any integer m, the primality of m can be determined by dividing m by all the primes up to the square root of m. This is because, instead of dividing by every number, you only have to divide by primes, because any other number is already divisible by a prime. You only have to check up to the square root of m, because anything beyond that would be multiplied by a number that was already checked. For example, the number 91 isn't divisible by 2, 3, and 5. But it is divisible by the last prime you have to check, 7. You could keep going, and find that the number was divisible by 13, but this is unnecessary because 91/7 is 13. Therefore, you only need to check numbers up to the square root of m. Since it it obvious to see if a number is divisible by 2 or 5, the only primes after 10 end with either 1, 3, 7, or 9. Using this method, even numbers above 10,000 can be tested for primality by checking only 23 primes!

As the numbers get higher, the number of possible divisors for the prime increases and the number of primes decreases. As said above, from 1-100 there are 25 primes, but from 18800-18900, the same span, there are only 5 primes!

As well as measuring primes, we can also measure the gaps between primes. The first few gaps measure 1, 2, 2, 4, 2, 4, 2, 4, 6... As the primes get higher the gaps also increase. The average gap between primes from 1-100 is 4, found by dividing 100 (the span) by 25 (the number of primes), and the average gap between 100-200 is 4.719. By the time you reach 25000, the average gap is over 10. However, the average gap pails in comparison compared to the huge gaps that occur.

Even though the average gap form 1-100 is 4, the highest gap in that time is 8, between 89 and 97. The next time the record is broken is from 113-127 a 14 number gap! Note that the first gap of exactly 10 isn't until 139-149, so even if a gap is a first occurrence, it doesn't have to be a record gap. So although a gap of 18 arrives at 523 (523-541), there isn't a gap of 16 until 1831. Similarly, a gap of 34 occurs at 1327, and a gap of 32 does not until 5591. By 20,000 the highest prime gap is an amazing 52! For all the first occurrence prime gaps, see here.

For other types of primes, and some of the largest known primes, see here.

Inverses of the Hyper Operators

Before reading this post, please read the previous post "The Hyper Operators"

Note: Some of the content described below is purely speculative and many theories are my personal theories.

From the previous post, it is known that the hyper operators include addition, multiplication, exponentiation, and the hyper 4 operator. However, the other functions such as subtraction are not mentioned. The remaining functions are the inverses of the hyper operators. The inverse of the hyper 1 operator (addition) is subtraction. The inverse of the hyper 2 operator (multiplication) is division. These concepts are simple. However, beyond this, things get more tricky.

The hyper 3 operator is exponentiation. The inverse of exponentiation is known as the logarithm. The logarithm consists of three parts: the base, the argument, and the solution. The standard notation for the logarithm is

logbx=y (pronounced log base b of x equals y)

A log without a base implies base 10.

The equation above implies that b^y=x. Therefore, log28=3 because 2^3=8. Also, logs provide solutions to equations such as 7^x=9 which has no rational solution. Logarithms also have many other properties. The natural logarithm ln(x) is a logarithm with base e (e=2.71828...).

Although I could go on listing the many properties of logs, the point of this post is to introduce a function of my own design, the hyperlogarithm. The hyperlogarithm (written hyperlog(x)) is the inverse of the upper hyper 4 operator. As with logarithms, hyperlogbx=y can only be true if b^^y=x Also similar to logs is that neither a log or a hyperlog can have a negative argument. Just as the hyper 4 operator can be extended to all real numbers, the hyperlog can too (provided that x>0 and b>0 and b is not equal to 1).

Similar to this, one can continue with hyperhyperlogarithms (the inverse of the hyper 5 operator) and so on. This brief post merely outlines the possibilities of the hyperlog and it may have other undiscovered properties.

The Hyper Operators

Everyone knows about the simple operation of addition. And it is also simple that multiplication is merely repeated addition (3*4=3+3+3+3). It is also simple that exponentiation is simply repeated multiplication e.g. 3^4=3*3*3*3. But less people know about the function of repeated exponentiation. This function is called the hyper 4 operator. The hyper 4 operator can be written as x^^y. x^^y equals x^x^x^x... with y x's.

Note: In this post the notation (^^) will represent the hyper 4 operator. Similar notation has be used for other functions. Please do not confuse them.

The hyper 4 operator has two types: lower and upper. The only difference between the lower and higher hyper 4 operators is the placement of the parentheses. For example, 3^^4, with the lower hyper 4 operator equals (3^3)^3)^3=7625597484987 while 3^^4 with the upper hyper 4 operator equals 3^(3^(3^3) which has over a trillion digits! As you can see, the upper hyper 4 operator produces much higher numbers than the lower hyper 4 operator. In this post, assume that all hyper 4 operators are upper, unless otherwise specified.

The hyper 4 operator can also be applied to non-integers. For example 3.5^^5=3.5^3.5^3.5^3.5^3.5. The real difficulty is repeating an exponent fractional times like in 3^^3.5. There is a simple solution to this, however. The solution is 3^(3^(3^(3^.5). Notice that the fourth three isn't whole, but taken to the .5 power. The 3^.5 represents half of a three. Another example is 5^^2.84. The solution is 5^(5^(5^.84) with 5^.84 representing .84 of a three. Even 3^^π can be calculated using an approximation of pi like 3.14 or 3.1415. For more info on pi see here. There are alternative approaches to the generalization of the hyper 4 operator. For more details see the comments below this post.

The hyper 4 operator can be applied to negative numbers as well. As with the previous example, having a negative number for x (in the equation x^^y) poses no issue (-2^^3=-2^(-2^-2), but for y, there is a real dilemma. How can there be a power tower with a negative number of elements in the tower? Just as 3^.5 can represent half a three, 3^-1 can represent a negative three. Using this rule, 3^^-2 would equal (3^-1)^(3^-1) or simply (1/3)^(1/3). Therefore, if x and y are positive numbers, x^^-y equals 1/x^(1/x^...1/x) with y 1/x's.

The rules above can be used to calculate the hyper 4 operator with any two real numbers.

If all functions are named this way, addition would be the hyper 1 operator, multiplication would be the hyper 2 operator, and exponentiation would be the hyper 3 operator. But what about higher operators like the hyper 5 operator? We can create the notation x^^^y for the hyper 5 operator. But what about other operators? How can 4^^^5.6 or even 8^^^^^^7 be solved? Using the rule of hyper operators, the hyper 5 operator must be repeated hyper 4 operators. As with the previous example, there are lower and upper hyper 5 operators. For example, for the lower hyper 5 operator, 3^^^4 would equal (3^^3)^^3)^^3, and the higher hyper 5 operator would be 3^^^4=3^^(3^^(3^^3). Beyond that, the hyper 6 operator would be repeated hyper 5 operators and so on. Higher hyper operators can also be written x^(y)^z with y being the number of karats used (2 karats for the hyper 4 operator, 3 for the hyper 5 operator etc.)

Many of these expressions are much to high for number form and must be expressed in scientific notation. But for things like 3^^65, the power of ten (e.g. x*10^y) will be too high to represent. In this case, a number can be represented as 10^(x*10^y) or even 10^(10^(10^(x*10^y). As you can see, these resemble hyper 4 operators themselves, except in base 10. All numbers can be represented this way.

Although the extension of the hyper 4 operator into all real numbers was my personal theory, the basis of this function and other info about huge numbers can be found here.

This notation leads to a whole new realm of incomprehensibly high numbers which push the limit of infinity.