## Prime Notation: Contents

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## What is Prime Notation?

In calculus, **prime notation ** (also called *Lagrange notation*) is a type of notation for derivatives. The “prime” is a single tick mark (a “prime”) placed after the function symbol, f.

For example: The function f′(x) is read “**f-prime of x.**”

Higher order derivatives are represented by adding more primes. For example, the third derivative of y with respect to x would be written as y′′′(x). You could, technically, keep adding primes, but most people switch to numerals. For example, the fourth derivative could be written as:

y′′′′(x) = f^{(4)}(x) = f^{(iv)}(x).

Lagrange (1736-1813) first introduced prime notation, but Lagrange prime notation isn’t the only way to symbolize a derivative. Another popular notation is Leibniz notation, which uses the capital Greek letter delta (Δ). Other notations that aren’t used as often include *Newton’s Notation* and *Euler’s notation*.

## Prime vs Apostophe

Prime notation because it uses the prime symbol (′), which are used to designate units and for other purposes. The prime symbol is commonly used, for example, to designate feet. Don’t confuse it with an apostrophe (′), as they are different characters. Some fonts make it very hard to discern though, and in for most intents and purposes, you can probably get away with just using an apostrophe (except when you’re publishing a paper or book).

## Other Meanings for Prime Notation

The prime symbol(‘) isn’t exclusively used in differential calculus. Another common use is in geometry, where it is used to denote **two distinct, yet similar objects** (e.g., vertices B and B’).

The term “Prime Notation” is also sometimes used to mean **grouping identical prime numbers** together to represent a number. For example, you could write the number 100 as 2^{2} x 5^{5}. This is a completely different meaning from the prime notation discussed here.

## Prime Number Theorem

The Prime Number Theorem helps to calculate probabilities of prime numbers in larger sets. It gives an **approximate number of primes less than or equal to any positive real number x.**

The theorem states that the “density” of prime numbers in the interval from 1 to x is approximately 1 / ln[x].

The following image shows how the approximation works. The black line is the actual density of primes from 0 to 200. For example, if you look at 40 on the chart, the density is 0.3. This is because there are 12 primes up to x = 40: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31 and 37. So the **actual density** is 12/40 = 0.3. The approximation 1/ln(40) = 0.27, which is a **reasonable approximation**, albeit a little low. The red line in the graph shows an alternate approximation 1/(ln(40)-1) = .37 — which is a little on the high side.

## Consequence of the Prime Number Theorem

An interesting **consequence of the prime number theorem **is that the average distance between consecutive primes (in the vicinity of *n*) is the logarithm of *n* (Robbins, 2006). For example, let’s say you took the interval 900 to 1100 (centered at 1,000). There are 30 prime numbers in the interval 900 to 1100.

The total distances are given in parentheses between each pair:

907 (4) 911 (8) 919 (10) 929 (8) 937 (4) 941 (6) 947 (6) 953 (14) 967 (4) 971 (6) 977 (6) 983 (8) 991 (6) 997 (12) 1009 (4) 1013 (6) 1019 (2) 1021 (10) 1031 (2) 1033 (6) 1039 (10) 1049 (2) 1051 (10) 1061 (2) 1063 (6) 1069 (18) 1087 (4) 1091 (2) 1093 (4) 1097

The average distance is 6.55:

(4 + 8 + 10 + 8 + 4 + 6 + 6 + 14 + 4 + 6 + 6 + 8 + 6 + 12 + 4 + 6 + 2 + 10 + 2 + 6 + 10 + 2 + 10 + 2 + 6 + 18 + 4 + 2 + 4) / 29 = 6.55.

Log(1000) gives us about 6.9. That’s a reasonable approximation, which gets a little better for a greater interval; The average distance from 800 to 1200 is 6.8.

## Ramanujan prime

A Ramanujan prime is another theory about the number of primes between certain points. The *n*th Ramanujan prime p is the smallest prime such that there are at least n primes between x and 2x. This is true for any x such that 2x > p.

### References:

Feldman, P. Prime Numbers and their statistical properties. Retrieved September 16, 2017 from: http://phillipmfeldman.org/primes/primes.html

Great Internet Mersenne Prime Search (GIMPS). Retrieved September 17, 2017 from: http://www.mersenne.org/.

Ribenboim, P. (1995). The Little Book of Big Primes. Springer Verlag.

Robbins, N. (2006). Beginning Number Theory. Jones and Bartlett Learning.

Vaughan, R. (1990). Harald Cramér and the distribution of prime numbers. Retrieved September 16, 2017 from: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.129.6847

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