Sylvester sequence: a(0)=2, a(n) = a(0)·a(1)·...·a(n-1) + 1. Equivalently: a(n) = a(n-1)² − a(n-1) + 1. First terms: 2, 3, 7, 43, 1807, 3263443... Grows like 2^(2^n). All terms pairwise coprime — Euclid's proof of infinite primes!
How to Use
Enter n:
- a(n): The n-th term
- Product: Running product
- Growth: Doubly exponential
Euclid's Proof
Since a(n) = product of all previous + 1, a(n) shares no prime factor with any previous term. Each term introduces at least one new prime, proving infinitely many primes exist.
Egyptian Fraction
1 = 1/2 + 1/3 + 1/7 + 1/43 + 1/1807 + ... (reciprocals of Sylvester terms). This is the greedy Egyptian fraction of 1. The sum converges to exactly 1 in the limit.
Step-by-Step Instructions
- 1Enter n.
- 2View a(n).
- 3See product.
- 4Check growth.
- 5Explore Egyptian fraction.