Partitions of n into x + y with x(x+1)/2 + y^2 prime

The question

conjecture_1. Conjecture: $a(n)>0$ for all $n>1$.

conjecture_2. We also conjecture that any integer $n>1$ can be written as $x+y$ ($x,y>0$) with $p_k(x)+p_{k+1}(y)$ prime, if and only if $k$ is among $3,39,99$.

conjecture_3. The pair $(k,m)$ works if $k$ is among $3,4,6$, and $m>k$ is not congruent to $k$ modulo $2$, in the sense that all sufficiently large integers $n$ can be written as $x+y$ ($x,y>0$) with $p_k(x)+p_m(y)$ prime.

• _Zhi-Wei Sun_

conjecture_4. For $k=5$, the pair $(5,m)$ works (with $m>5$) if $m$ is congruent to $0$ or $4$ modulo $6$, in the sense that all sufficiently large integers $n$ can be written as $x+y$ ($x,y>0$) with $p_5(x)+p_m(y)$ prime.

• _Zhi-Wei Sun_

conjecture_5. The only pairs $(k,m)$ with $2<k\le 10$ and $k<m\le 100$ such that any integer $n>1$ can be written as $x+y$ ($x,y>0$) with $p_k(x)+p_m(y)$ prime, are the $31$ pairs in sunPolygonalPairs.

• _Zhi-Wei Sun_

Number of ways to write $n=x+y$ ($x,y>0$) with $x(x+1)/2+y^2$ prime.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.OEIS.«228425» (5 statements).

theorem conjecture_1 (n : ℕ) (hn : n > 1) : a n > 0

theorem conjecture_2 (k : ℕ) (hk : 3 ≤ k) :
    (∀ n : ℕ, 1 < n → ∃ x y : ℕ, 0 < x ∧ 0 < y ∧ x + y = n ∧
      Nat.Prime (Nat.polygonalNumber k x + Nat.polygonalNumber (k + 1) y)) ↔
    k = 3 ∨ k = 39 ∨ k = 99

theorem conjecture_3 (k m : ℕ) (hk : k = 3 ∨ k = 4 ∨ k = 6) (hm : k < m)
    (hmod : ¬ m ≡ k [MOD 2]) :
    ∀ᶠ (n : ℕ) in Filter.atTop, ∃ x y : ℕ, 0 < x ∧ 0 < y ∧ x + y = n ∧
      Nat.Prime (Nat.polygonalNumber k x + Nat.polygonalNumber m y)

theorem conjecture_4 (m : ℕ) (hm : 5 < m) (hmod : m ≡ 0 [MOD 6] ∨ m ≡ 4 [MOD 6]) :
    ∀ᶠ (n : ℕ) in Filter.atTop, ∃ x y : ℕ, 0 < x ∧ 0 < y ∧ x + y = n ∧
      Nat.Prime (Nat.polygonalNumber 5 x + Nat.polygonalNumber m y)

theorem conjecture_5 (k m : ℕ) (hk : 2 < k) (hk10 : k ≤ 10) (hm : k < m) (hm100 : m ≤ 100) :
    (∀ n : ℕ, 1 < n → ∃ x y : ℕ, 0 < x ∧ 0 < y ∧ x + y = n ∧
      Nat.Prime (Nat.polygonalNumber k x + Nat.polygonalNumber m y)) ↔
    (k, m) ∈ sunPolygonalPairs

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

• A228425

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.