claimroutineinferred_dep

Finite multiplier set and initial residue class

Picks finite multiplier set K\mathcal K and residue class for 13 forms
starting point for the prime-pattern count
Claim·§6.2 A finite multiplier set and the initial residue conditions·inside lemPrime patterns with controlled reduction orders

Statement (paper)

There is a finite set K⊂Z>0\mathcal K\subset\Z_{>0}, fixed before AA, LL and XX, such that for every A∈ZA\in\Z there exist a modulus W0W_0 (ranging in a fixed finite set as AA varies), integers kν∈Kk_\nu\in\mathcal K (1≤ν≤t=13)(1\le\nu\le t=13), signs ϵν∈{1,−1}\epsilon_\nu\in\{1,-1\}, a bounded open box B′\mathcal B' and a vector class modulo W0W_0 such that on this class all Ψν(T)=ϵνFν(T)/kν\Psi_\nu(T)=\epsilon_\nu F_\nu(T)/k_\nu are integers, nonzero modulo every prime dividing W0W_0, with Ψν≡1(mod8)\Psi_\nu\equiv1\pmod 8, Ψν≡3(mod7)\Psi_\nu\equiv3\pmod 7, and Ψν≢−1\Psi_\nu\not\equiv-1 modulo every odd prime dividing W0W_0; moreover W0/kνW_0/k_\nu is divisible by every prime dividing W0W_0, and there are constants 0<cA<CA0<c_A<C_A with cAX≤Ψν(T)≤CAXc_AX\le\Psi_\nu(T)\le C_AX for all T∈XB′T\in X\mathcal B' and 1≤ν≤t1\le\nu\le t.

In plain English

There is a finite set K\mathcal K of positive integers, chosen before AA, LL, XX, such that for each AA one can pick a modulus W0W_0 (from a finite set), multipliers kν∈Kk_\nu\in\mathcal K, signs ϵν\epsilon_\nu, a bounded open box B′\mathcal B' and a residue class mod W0W_0 on which the 13 forms Ψν=ϵνFν/kν\Psi_\nu=\epsilon_\nu F_\nu/k_\nu are integers, nonzero mod each prime dividing W0W_0, congruent to 11 mod 88 and 33 mod 77, and not −1-1 mod any odd prime dividing W0W_0. Also W0/kνW_0/k_\nu is divisible by every prime dividing W0W_0, and all Ψν(T)\Psi_\nu(T) lie between cAXc_AX and CAXC_AX for T∈XB′T\in X\mathcal B'.

Why it's here

This sets up the starting point for a prime-pattern argument: thirteen linear forms in TT that are integer-valued, positive and of size about XX on a box, with congruence conditions built in. The conditions Ψν≡1(mod8)\Psi_\nu\equiv1\pmod 8 and 3(mod7)3\pmod 7 are what the previous claim needs for the orders r+1±2ar+1\pm2a. The residue conditions mod odd primes of W0W_0 keep Ψν\Psi_\nu away from −1-1 there. The next claim can then apply a Green–Tao–Ziegler count to find TT where all Ψν(T)\Psi_\nu(T) are prime, so that ϵνkνΨν\epsilon_\nu k_\nu\Psi_\nu gives the required numbers ϵkr\epsilon kr.
Classified routine:
A local-conditions setup in the usual style of Green–Tao–Ziegler applications: CRT bookkeeping that picks the modulus, signs and multipliers so the linear forms are integral, positive, of size about X and in the right residue classes. It adds no new idea beyond the preceding claims.

Uses

claimReduction orders r+1±2a and common factors

Used by

claimGreen-Tao-Ziegler lower bound for prime patterns

Sources

PaperOpen
Section
§6.2 A finite multiplier set and the initial residue conditions
Page
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sections/06-primes.tex:184
Leannot aligned

Phase 2 aligns this node to a Lean declaration.

Friction

  • info
    inferred_dep
    outline: "Finite multiplier set and initial residue class" rests on claim-pri-pattern-1