What mod 6 means
Looking at numbers mod 6 means grouping them by their remainder after division by 6. Every integer falls into one of six residue classes: 0, 1, 2, 3, 4, or 5 mod 6.
Why this helps
You do not need a long lesson in modular arithmetic to get the basic idea. This page gives the shortest useful explanation, then points you back to the visual and analytical views where the pattern becomes easier to see.
Why primes narrow to two classes
Any number congruent to 0, 2, or 4 mod 6 is even, so it cannot be prime unless it is 2. Any number congruent to 3 mod 6 is divisible by 3, so it cannot be prime unless it is 3. For primes greater than 3, that leaves only residue classes 1 and 5 mod 6.
Why this matters for twin primes
A typical twin-prime pair above (3, 5) looks like (6k - 1, 6k + 1). That means mod 6 quickly reveals why the pair members and the center between them keep falling into a narrow pattern.
Why this is a filter rather than a proof
The mod-6 pattern tells you where prime candidates can survive once divisibility by 2 and 3 is removed. It does not tell you that every number in those residue classes is prime. Numbers such as 25 and 35 sit in allowed residue classes and are still composite. That is why modular structure helps narrow the search without settling it.
Why twin centers make the same pattern easier to see
If a twin-prime pair looks like (6k - 1, 6k + 1), then the number in the middle is 6k. That is why twin centers so often land on multiples of 6. The center compresses the pair into one visual anchor, which makes the same modular story easier to scan in the Lab and easier to summarize in Analysis.
Why the site highlights it
Mod 6 is one of the fastest ways to move from raw numbers to visible structure. The Lab uses it as a visual mode, Analysis uses it as a structural read, and the Glossary keeps the key terms short when you do not want a longer explanation.
How to read this pattern on the site
Use the Lab when you want the quickest visual impression of how residue classes narrow the field. Use Analysis when you want counts and summaries for the same structure. Use the Glossary when you only need short definitions for modulus, residue class, or arithmetic progression instead of a fuller article.
Where to go next
Use these links to keep reading or jump back into the live number views.
See it in the Lab
Use the Mod 6 view to see the residue pattern rather than only reading about it.
Read the modular interpretation
Analysis summarizes the pair and center residue counts across the selected range, so you can compare the article idea with live data.
Keep the terms nearby
Use the Glossary for Mod 6, residue class, and arithmetic progression definitions.
Connect the pattern to candidate finding
The finding guide shows how residue-class filters help you search for likely twin-prime candidates.