Alphabeta Math
CounterexampleConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-05
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A noncoprime residue class has no Dirichlet conclusion

Statement refuted

The coprimality hypothesis (a,q)=1 in Dirichlet's theorem cannot be dropped.

Facts & Assumptions

Given: Dirichlet's theorem applies only to reduced residue classes (Dirichlet's theorem on primes in arithmetic progressions).

Counterexample

technique · direct
1.1

Take q=6 and a=3. Every integer congruent to 3 modulo 6 is 6m+3=3(2m+1), so it is divisible by 3.

givenalgebra
2.1

Hence the only prime in that residue class is 3 itself. In particular, there are not infinitely many such primes. So the reduced-residue hypothesis in Dirichlet's theorem on primes in arithmetic progressions is indispensable.

step 1.1givenalgebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

2 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources