Facts about Prime Numbers
- 11
Fermat numbers, expressed as 2^(2^n) + 1, have only five known primes among them: F0=3, F1=5, F2=17, F3=257, and F4=65537, while all tested larger Fermat numbers are composite despite intensive computational searches.
- 10
Sophie Germain proved in 1819 that if p is an odd prime and 2p + 1 is also prime, then Fermat's Last Theorem holds for that exponent, creating a new class of primes now bearing her name.
- 09
Bertrand's postulate, proven by Chebyshev in 1852, guarantees that for every integer n greater than 1, at least one prime exists between n and 2n, ensuring primes never have arbitrarily large gaps.
- 08
Eratosthenes developed a sieve algorithm around 240 BCE that efficiently identifies all prime numbers up to any given limit by systematically eliminating multiples of each prime in sequence.
- 07
Goldbach's conjecture, proposed in 1742, asserts that every even number greater than 2 can be expressed as the sum of two primes, remaining unproven despite verification for numbers exceeding four quintillion.
- 06
Twin primes like 11 and 13 or 29 and 31 appear mysteriously throughout the number line, yet mathematicians remain unable to prove whether infinitely many such pairs exist, despite centuries of effort.
- 05
Every even number greater than 2 fails to be prime because it shares 2 as a divisor, making odd numbers the exclusive domain where all primes except 2 can possibly exist.
- 04
Among 1 and 100, exactly 25 prime numbers exist, a density that decreases as numbers grow larger, following the Prime Number Theorem proven rigorously by Hadamard and de la Vallée Poussin in 1896.
- 03
Euclid proved around 300 BCE that infinitely many primes exist by demonstrating that no finite list of primes can be complete, establishing mathematics' first proof by contradiction.
- 02
The largest known prime number, discovered in December 2018, contains 24,862,048 digits and is designated M82589933, part of the Mersenne prime family.
- 01
In 1978, RSA encryption was invented using two 64-digit prime numbers as its mathematical foundation, revolutionizing digital security.