Interactive mathematics laboratory
Primes in arithmetic progression
Can we reach every prime starting from 3 and 5?
Three numbers a, a+d, a+2d form an arithmetic progression. Here all three must be prime: for example 3, 5, 7; 3, 7, 11; 5, 11, 17.
We begin with S₀ = {3, 5}. This computational check does not prove a general conjecture.
In 1993 Siemion Fajtlowicz conjectured that every odd prime belongs to a three-prime arithmetic progression. Mode B explores a stronger question: can we reach them all iteratively starting from {3,5}? In mode A the answer is already no: 13 is missing. Historical source: Written on the Wall, no. 783.
Set up the experiment
The default pair is 3 and 5. You may choose two distinct odd primes for a new experiment. After a complete closure, “Raise limit and continue” keeps every prime already found while this page remains open; if you leave the limit unchanged, it raises it automatically.
Choose the parameters and start the experiment.
Maximum N: 5,000,000; maximum M: 25,000,000. Computation runs in your browser. Each phase has a budget of 50 million pairs; if exhausted, the result is partial.
Two rules, two different questions
A — Forward generation
Given two available primes a < b, mode A tests c = 2b − a as the third term. From (3,5) we obtain 7, then from (3,7) we obtain 11.
B — Full closure
Mode B also tests c = 2a − b as the first term and c = (a+b)/2 as the middle term. Only odd primes greater than 2 are accepted.
With M > N we may temporarily use primes beyond N while measuring coverage only up to N.
Results
Start an experiment to see the results.
What happened in the latest generation?
From (3,5) we obtain 7 because 2·5 − 3 = 7 is prime.
New primes by generation
Coverage up to the examined value
Generations
Generation 0: 3, 5.