Delta_min

Number theory research program by Ken Clements (March 8, 2026).

Overview

This program investigates the prime-complete property of consecutive integer products n(n+1). For each n, it computes:

• omega — the number of distinct prime divisors of n(n+1)
• Pidx — the prime index of the greatest prime divisor of n(n+1)
• delta = Pidx − omega — the count of "missing" prime divisors between 2 and the greatest prime divisor

When delta = 0, the product n(n+1) has no missing prime divisors in the sequence from 2 up to its greatest prime factor — i.e., the number is prime-complete.

The search is conducted over doubling intervals of powers of 2, tracking the minimum delta, maximum omega, and minimum Pidx found in each interval.

Algorithm

Because n and n+1 are always coprime:

• omega(n(n+1)) = omega(n) + omega(n+1)
• Pidx(n(n+1)) = max(Pidx(n), Pidx(n+1))

The loop recycles omega(n+1) and Pidx(n+1) as the next omega(n) and Pidx(n) to avoid redundant calculations. Dynamic pruning cuts off factorization early when the result cannot improve the current best delta.

Parallelism

Uses OpenMP for multi-threaded parallel search with dynamic scheduling. Results from each thread are merged at the end of each interval.

Build

gcc-15 -O3 -march=native -fopenmp -o Delta_min Delta_min.c -lm

Usage

./Delta_min [interval_start] [interval_initial_size] [number_of_intervals] [max_Pidx]

Defaults: start=1, initial size=1, 40 intervals, max Pidx=30

Example:

./Delta_min 1 1 40 30

Output

For each interval, the program prints:

• Interval start value
• Minimum delta found (and the n, Pidx, omega at that minimum)
• Maximum omega found (and the n)
• Minimum Pidx found (and the n)
• Wall-clock time for the interval

A CSV file delta_min_data.csv is also written with the full results.

Background

The primorial p_k# is the product of the first k primes. With max_pidx = 30, the program can correctly cover n(n+1) up to p_30#:

31,610,054,640,417,607,788,145,206,291,543,662,493,274,686,990

License

MIT License — see LICENSE.