Primorials¶
The primorial of a prime \(p_k\) is
(the product of the first \(k\) primes). Primorials are a natural “smoothing knob” in experimental number theory: they amplify small prime structure and appear in constructions like Euclid numbers \(p_k\# \pm 1\). [Prime Pages (UTM), 2025, The OEIS Foundation Inc., 2025]
Key facts¶
Growth: \(\log(p_k\#) = \sum_{i\le k} \log p_i\); by the prime number theorem this is asymptotic to \(p_k\) (in a coarse sense). [Hardy and Wright, 2008]
Euclid numbers: \(p_k\# + 1\) is coprime to all primes \(\le p_k\), but is usually composite (so: “Euclid’s proof does not generate primes”). [Prime Pages (UTM), 2025]
What to experiment with¶
Euclid-number factorization: For \(E_k = p_k\# \pm 1\), try to find small factors and compare +1 vs. -1.
Primorial wheels: Use primorials to build “wheel sieves” and benchmark against a simple sieve.
Primorial primes / plus/minus primes: Scan for primes of the form \(p_k\# \pm 1\) and visualize rarity.
References¶
See Prime Pages (UTM) [2025], The OEIS Foundation Inc. [2025], Wikipedia contributors [2025].