A prime number is a natural number greater than 1 that has exactly two distinct positive divisors: 1 and itself. In other words, it cannot be formed by multiplying two smaller natural numbers. The first few primes are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, etc. Note that 2 is the only even prime number — all other even numbers are divisible by 2 and therefore composite.

Our tool takes your starting integer and incrementally checks each subsequent number using an optimized trial division algorithm (the 6k±1 method). For each candidate number n, it checks divisibility only by numbers of the form 6k±1 up to √n, skipping all even numbers and multiples of 3. This makes it approximately 3 times faster than naive trial division. You can choose to find just the next prime, or generate a sequence of up to 50 consecutive primes.

The first prime number is 2. It is also the only even prime. The number 1 is NOT prime — by definition, a prime must have exactly two distinct positive divisors (1 and itself), but 1 only has one divisor (itself). The sequence of primes begins: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53...

Yes! This was proven by the ancient Greek mathematician Euclid around 300 BCE. His elegant proof shows that if you multiply all known primes together and add 1, the result is either a new prime or has a prime factor not in the original list. This means no finite list can contain all primes — there are infinitely many. The largest known prime (as of 2024) is 2^82,589,933 − 1, a Mersenne prime with over 24.8 million digits.

A prime gap is the difference between two consecutive prime numbers. For example, between 7 and 11, the gap is 4. While small gaps (like 2 for twin primes) are common, gaps can be arbitrarily large — you can find a sequence of any length containing no primes at all. The average gap near a number n is approximately ln(n) (the natural logarithm). Our tool displays the gap for each prime found, helping you analyze the distribution of primes.

Prime numbers are the building blocks of all integers (via the Fundamental Theorem of Arithmetic — every integer > 1 factors uniquely into primes). Beyond pure mathematics, primes are critical in:

• Cryptography: RSA encryption and many other systems rely on the difficulty of factoring large semiprimes (products of two large primes).
• Hash functions: Prime moduli help distribute hash values uniformly.
• Computer science: Pseudorandom number generation, error-correcting codes, and algorithm design.
• Nature: Cicadas emerge in prime-numbered year cycles (13 or 17 years) to avoid predators with periodic life cycles.

The Sieve of Eratosthenes is one of the oldest known algorithms for finding all primes up to a given limit. It works by iteratively marking the multiples of each prime starting from 2. While extremely efficient for generating all primes up to a bound, it requires O(n) memory. For finding the next prime after a specific number, direct primality testing (as used in our tool) is more practical and memory-efficient.

For small numbers, use trial division: check divisibility by all integers from 2 up to √n. If none divide evenly, n is prime. For example, to check if 97 is prime: √97 ≈ 9.85, so test divisors 2, 3, 4, 5, 6, 7, 8, 9. Since none divide 97, it is prime. For larger numbers, more sophisticated tests like the Miller-Rabin primality test or the AKS primality test are used. Our tool handles all the heavy lifting for you!

Twin primes are pairs of primes that differ by exactly 2 — the smallest possible gap between odd primes. Examples include (3, 5), (5, 7), (11, 13), (17, 19), (29, 31), and (41, 43). It is conjectured (but not yet proven) that there are infinitely many twin primes — this is the famous Twin Prime Conjecture. Using our tool, you can easily spot twin primes by looking for a gap of 2 between consecutive results.

A Mersenne prime is a prime number of the form 2^p − 1, where p itself is prime. Named after the French monk Marin Mersenne, these primes are central to the search for large primes because there are efficient primality tests specifically for numbers of this form (the Lucas-Lehmer test). As of 2024, the largest known prime is 2^82,589,933 − 1, a Mersenne prime discovered in December 2018 by the Great Internet Mersenne Prime Search (GIMPS).