LemmaStatement: AI-adaptedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-13

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There are arbitrarily large primes congruent to $2$ modulo $3$

Statement

For every natural $M$ , there is a prime $p > M$ with $p \equiv 2 (m o d 3)$ .

Facts & Assumptions

Given: A natural number $M$ .

[L1]

Every positive integer has a finite prime factorisation, including multiplicities; $1$ is represented by the empty product (Every integer $n \geq 1$ is a finite product of primes: there are $r \in N$ and a list $p : r \to Z$ of primes with $n = \prod_{i < r} p_{i}$ , the case $n = 1$ being the empty product).

[L2]

Congruence modulo $3$ and multiplication of residue classes obey integer modular algebra (Congruence modulo an integer: $a \equiv b (m o d n)$ when $n ∣ (a - b)$ , including the moduli $0$ and $1$ , For every natural $n$ , $(Z / n, +)$ is an abelian group, multiplication is a commutative monoid operation, and both distributive laws hold).

[L3]

A finite product in a monoid is $1$ on the empty family and obeys the product recursion (The product $g_{0} g_{1} \dots g_{n - 1}$ of a finite list in a monoid, by recursion, with the empty product ( $n = 0$ ) equal to the identity).

Proof

technique · constructive

1.1

Let $q_{1}, \dots, q_{r}$ be all primes at most $M$ that are congruent to $2$ modulo $3$ , allowing $r = 0$ , and put $N = 3 q_{1} \dots q_{r} - 1$ . Then $N \equiv 2 (m o d 3)$ and $N > 1$ ; when the list is empty, $N = 2$ .

L2L3construct

2.1

No $q_{i}$ divides $N$ , because $N \equiv - 1 (m o d q_{i})$ . Also $3$ does not divide $N$ .

step 1.1L2

3.1

In a prime factorisation of the positive integer $N$ , not every factor can be congruent to $1$ modulo $3$ , since their product is congruent to $2$ ; no factor is congruent to $0$ by step 2.1, so some prime factor $p$ is congruent to $2$ modulo $3$ .

step 1.1step 2.1L1L2

4.1

That $p$ is not among the $q_{i}$ by step 2.1, hence $p > M$ . It is the required prime.

step 2.1step 3.1discharge-construct∎

Depends on

Used by

Every nonempty finite set of n nonzero integers has a sum-free subset of size greater than n/3 Theorem

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 68 results over 23 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

M. Bucic, Probabilistic Method, proof of Theorem 2.3 (standard reference, not scraped)

There are arbitrarily large primes congruent to 22 modulo 33