claimroutineinferred_dep

Surviving prime pattern satisfies order conditions

Shows some prime pattern has gcd⁡(d1,d2)∣4\gcd(d_1,d_2)\mid4 and r∤d1d2r\nmid d_1d_2
finishes the prime-pattern lemma
Claim·§6.7 Completing the prime-pattern construction·inside lemPrime patterns with controlled reduction orders

Statement (paper)

Let LL and XX be chosen as above. Then at least one prime pattern survives, namely one with no common odd factor l>Ll>L in any of its pairs of reduction orders. For such a pattern, the reduction orders d1,d2d_1,d_2 at a place of FF above rr satisfy gcd⁡(d1,d2)∣4\gcd(d_1,d_2)\mid 4 and r∤d1d2r\nmid d_1d_2, and all the primes involved exceed the prescribed BB.

In plain English

For suitable LL and XX, at least one prime pattern has no common odd factor l>Ll>L in any of its pairs of reduction orders. For such a pattern, the reduction orders d1,d2d_1,d_2 at a place above rr satisfy gcd⁡(d1,d2)∣4\gcd(d_1,d_2)\mid 4 and r∤d1d2r\nmid d_1d_2, and all the primes exceed BB.

Why it's here

This finishes the proof of the prime-pattern lemma. The Green–Tao–Ziegler lower bound gives about X3/(log⁡X)tX^3/(\log X)^t patterns with only a (log⁡L)−C0(\log L)^{-C_0} loss, while the uniform upper sieve and the count of primes with a large common order factor show that the bad patterns, with a common odd factor above LL, number O(1/L)O(1/L) times that. So for large LL a pattern survives. The order-structure claim then shows that the 2-part of the gcd is exactly 4, that no odd common factor remains, and that rr divides neither order, which gives the stated conditions.
Classified routine:
A direct combination of the earlier claims: the Green–Tao–Ziegler lower bound (claim-pri-pattern-3) is compared against the bad-pattern count obtained from the upper sieve and the common-order-factor bound (claims 5 and 6), followed by the order-structure claim (claim-pri-pattern-1). This is standard counting bookkeeping.

Uses

claimGreen-Tao-Ziegler lower bound for prime patternsclaimFew primes with large common order factorclaimUniform upper sieve on affine fibersclaimReduction orders r+1±2a and common factors

Used by

Nothing in the paper

Sources

PaperOpen
Section
§6.7 Completing the prime-pattern construction
Page
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Label
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Line
sections/06-primes.tex:670
Leannot aligned

Phase 2 aligns this node to a Lean declaration.

Friction

  • info
    inferred_dep
    outline: "Surviving prime pattern satisfies order conditions" rests on claim-pri-pattern-3
  • info
    inferred_dep
    outline: "Surviving prime pattern satisfies order conditions" rests on claim-pri-pattern-5
  • info
    inferred_dep
    outline: "Surviving prime pattern satisfies order conditions" rests on claim-pri-pattern-6
  • info
    inferred_dep
    outline: "Surviving prime pattern satisfies order conditions" rests on claim-pri-pattern-1