Finding Twin Primes
Finding a twin-prime pair is straightforward in small ranges: look for two prime numbers that differ by exactly 2. The deeper challenge is learning which candidates are worth checking, which shortcuts are only filters, and how to move from quick spotting to real inspection.
This page is about procedure rather than proof. It shows how to recognize twin-prime candidates, what quick filters help, where false candidates come from, and how the site's tools let you move from a simple search idea to richer structural inspection.
A twin-prime pair is a pair of primes with a gap of 2. So the simplest search method is: take a prime p, check whether p + 2 is also prime, and record the pair if it is. In small ranges this is enough to generate real examples quickly. If p equals 11 and p plus 2 equals 13, you have a twin-prime pair. If p equals 13 and p plus 2 equals 15, the pattern fails because 15 is composite.
In practice, people rarely test every whole number from scratch. They work from a list of primes or from candidates that have already passed small divisibility filters. That is why residue classes such as 1 and 5 mod 6 matter so much: they eliminate many impossible cases before you do deeper checking. A practical workflow is often: narrow the candidates quickly, then inspect the survivors more carefully.
For primes greater than 3, only two residue classes mod 6 remain possible. That means a typical twin-prime pair above (3, 5) looks like (6k - 1, 6k + 1). This is a useful candidate pattern, not a proof that those numbers are prime. Modular filters tell you where to look, but primality still has to be checked. In other words, Mod 6 is a search shortcut, not a guarantee.
Many numbers fit the right-looking shape and still fail. A pair can land in the correct residue classes and still contain a composite number with a nontrivial factor. For example, (35, 37) has the right-looking gap of 2, but 35 is composite. So does (77, 79), where 77 is composite even though the pair shape looks promising at first glance. That is why it helps to treat modular structure as a fast screening tool rather than as a guarantee.
Finding examples is useful for building intuition, checking ranges, and understanding how twin-prime patterns appear in practice. But no matter how many examples you collect in finite ranges, you still have not proved that infinitely many twin-prime pairs exist. This is a good place to keep the difference between observing a pattern and proving an infinite statement clear.
Use Explorer when you want the most direct working surface: row-by-row values, prime roles, twin centers, and divisor details. Use the Lab when you want to scan visually first and notice likely clusters. Then move into Analysis if you want to understand how the examples you found fit into larger spacing or modular patterns. The Mod 6 page, prime-gaps page, and twin-prime page all help explain why the candidates you see behave the way they do.
Use these links to keep reading or jump back into the live number views.
Explorer is the clearest place to test candidate pairs and inspect neighborhood roles one row at a time.
Use the modular explainer if you want the residue filter idea in plain language first.
The twin-primes page gives the larger mathematical context around the exact pattern you are trying to find.
Analysis helps connect the candidate-finding process to larger gap and density structure.