Proth numbers: k·2^n+1 with k odd, k<2^n. Proth's theorem: P is prime if ∃a with a^((P-1)/2)≡−1(mod P). This is deterministic and fast! Used in the search for large primes. The largest known Proth prime has millions of digits.
How to Use
Enter k and n:
- Proth number: k·2^n+1
- Primality: Proth's test
- Sequence: List of Proths
Proth's Theorem
If P=k·2^n+1 with k<2^n, and a^((P-1)/2)≡−1(mod P) for some a, then P is prime. Common choices: a=3. This is remarkably efficient — O(n²) bit operations. No false positives possible!
Records
Many of the largest known primes are Proth primes. The distributed computing project PrimeGrid searches for large Proth primes. They're faster to test than general numbers because of Proth's efficient theorem.
Step-by-Step Instructions
- 1Enter k (odd).
- 2Enter n.
- 3Compute k·2^n+1.
- 4Test primality.
- 5View sequence.