Theorems and Conjecture related to Prime Numbers (original) (raw)
Last Updated : 23 Jul, 2025
Prime numbers are special numbers that can only be divided by 1 and themselves. Examples include 2, 3, 5, 7, and 11. Over the years, many interesting rules and patterns about prime numbers have been discovered. Some of these, like Euclid’s theorem, show that there are infinitely many primes, while others, like the Prime Number Theorem, explain how primes are spread out as numbers get bigger.
There are many such theorems, some of these interesting results are discussed below:
**Fundamental Theorem of Arithmetic
Every positive integer greater than 1 can be expressed as a product of prime numbers in a unique way, up to the order of the factors.
For example: 60 = 22 × 3 × 5
**Euclid's Theorem (Infinitude of Primes)
There are infinitely many prime numbers.
**Prime Number Theorem
The number of prime numbers less than or equal to n, denoted by π(n), asymptotically approaches \frac{n}{\ln n} i.e., \lim_{n \to \infty} \frac{\pi(n)}{n / \ln n} = 1.
**Bertrand’s Postulate (Chebyshev’s Theorem)
For any integer n > 1, there exists at least one prime number p such that n < p < 2n.
**Example: For n = 10, there is at least one prime between 10 and 20 (e.g., 11, 13, 17, 19).
**Wilson’s Theorem
A positive integer p is a prime number if and only if:
(p − 1)! ≡ −1 (mod p)
**For Example: For p = 5, 4! = 24 and 24 ≡ −1 (mod5).
**Goldbach's Conjecture (Unproven)
Every even integer greater than 2 can be expressed as the sum of two prime numbers.
**For Example: 4 = 2 + 2, 6 = 3 + 3, 8 = 3 + 5, etc.
**Fermat's Little Theorem
If p is a prime number and a is any integer such that a≢ 0 (mod p), then:
ap−1 ≡ 1 (mod p).
**Dirichlet’s Theorem on Primes in Arithmetic Progressions
If a and d are coprime integers, then the arithmetic progression a, a + d, a + 2d, . . . contains infinitely many prime numbers.
**Example: The sequence 5, 11, 17, 23,. . . (arithmetic progression with a = 5) contains infinitely many primes.
**Legendre’s Conjecture (Unproven)
There is always at least one prime number between n2 and (n + 1)2 for every positive integer n.
**Example: For n = 2, there is a prime (5) between 22 = 4 and (2 + 1)2 = 32 = 9.
**Twin Prime Conjecture (Unproven)
There are infinitely many twin primes, i.e., pairs of primes p and p + 2.
**Example: The pair (3, 5), (11, 13), and (17, 19) are all twin primes.
**Zsigmondy’s Theorem
For any integers a > b > 0, and for n > 1, the number an − bn has at least one prime factor that does not divide ak − bk for any k < n, except in certain known cases.
**For Example: Consider the expression 25 − 15 = 31. Here, 31 is a prime number and a **primitive prime factor of 25 − 15 because it does not divide any of the earlier terms, like 21 − 11 = 1, 22 − 12 = 3, or 23 − 13 = 7.
**Mertens' Theorem
The sum of the reciprocals of the prime numbers diverges. \sum_{p \, \text{prime}} \frac{1}{p} = \infty
Conclusion
Prime numbers are like the building blocks of all numbers, and the theorems about them help us understand how they work and where they appear. From Euclid’s proof that there are endless primes to the Prime Number Theorem explaining their spread, these theorems uncover important patterns.
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