Representations of n = p + q + (n bmod 2)q with Jacobi symbol constraints

The question

conjecture_1. Conjecture: $a(n)>0$ for all $n>5$.

• _Zhi-Wei Sun_, Dec 30 2012

conjecture_2. Refinement conjecture (odd case): Any odd number $2n+1>64$ not among $105,247,255,1105$ can be written as $p+2q$, where $p$ and $q$ are primes, and $\left(\frac{q}{p^{\prime }}\right)=1$ for any prime divisor $p^{\prime }$ of $2n+1$.

• _Zhi-Wei Sun_, Dec 30 2012

conjecture_3. Refinement conjecture (even case): Any even number $m=2n>8$ not among $32$ and $152$ can be written as $p+q$, where $p$ and $q\le m/2$ are primes, and $\left(\frac{(q+1)/2}{p^{\prime }}\right)=1$ for any prime divisor $p^{\prime }$ of $m+1$. (The OEIS comment writes $q\le n/2$ after renaming the argument of $a(\cdot )$ to $2n$, which is a slip for $q\le (2n)/2=n=m/2$, matching the bound in the definition of $a(2n)$ and yielding the stated exceptional set $\{32,152\}$.)

• _Zhi-Wei Sun_, Dec 30 2012

The sequence $a(n)$ counts the number of ways to write $n=p+q+(n\mathrm{mod}2)q$, where $p$ is an odd prime and $q\le n/2$ is a prime such that $\left(\frac{q}{n}\right)=1$ if $n$ is odd, and $\left(\frac{(q+1)/2}{n+1}\right)=1$ if $n$ is even.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.OEIS.«191004» (3 statements).

theorem conjecture_1 (n : ℕ) (hn : 5 < n) : 0 < a n

theorem conjecture_2 (m : ℕ) (hm : 64 < m) (hodd : Odd m)
    (hnot : m ∉ ({105, 247, 255, 1105} : Finset ℕ)) :
    ∃ p q : ℕ, p.Prime ∧ q.Prime ∧ m = p + 2 * q ∧
      ∀ p' ∈ m.primeFactors, jacobiSym (q : ℤ) p' = 1

theorem conjecture_3 (m : ℕ) (hm : 8 < m) (heven : Even m)
    (hnot : m ∉ ({32, 152} : Finset ℕ)) :
    ∃ p q : ℕ, p.Prime ∧ q.Prime ∧ m = p + q ∧ q ≤ m / 2 ∧
      ∀ p' ∈ (m + 1).primeFactors, jacobiSym (((q + 1) / 2) : ℤ) p' = 1

What counts as progress

• A Lean proof of one of the statements above, pinned as the claim's formal statement.
• Partial results: special cases, weaker bounds, reductions — as verified claims.
• Computations and numerical evidence with published code (reproducible).
• Literature: the problem may have been solved or partly solved already. Report it as a literature claim.
• A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

• A191004
• Conjectures involving primes and quadratic forms by Zhi-Wei Sun, arXiv:1211.1588 (2012)

Source and licence

Imported from Formal Conjectures (OEIS), commit 2f91d251c07f. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.