Ways to write n = x + y + z with six linear and quadratic forms prime

The question

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

conjecture_2. Moreover, any integer $n>6$ can be written as $x+y+z$ with $x$ among $3,4,6,10,15$ such that $2y-1,2z-1,2xy-1,2xz-1,2yz-1$ are prime.

conjecture_3a. Conjecture (i), part 1: Any integer $n>6$ can be written as $x+y+z$ ($x,y,z>0$) with $2x-1,2y-1,2z-1$ and $2xyz-1$ all prime and $x$ among $2,3,4$.

• _Zhi-Wei Sun_

conjecture_3b. Conjecture (i), part 2: Each integer $n>2$ can be written as $x+y+z$ ($x,y,z>0$) with $2x+1,2y+1,2z+1$ and $2xyz+1$ all prime and $x$ among $1,2,3$.

• _Zhi-Wei Sun_

conjecture_4. Conjecture (ii): Each integer $n>4$ can be written as $x+y+z$ with $x=3$ or $6$ such that $2y+1,2xyz-1$ and $2xyz+1$ are prime.

• _Zhi-Wei Sun_

conjecture_5a. Conjecture (iii), part 1: Every integer $n>5$ can be written as $x+y+z$ ($x,y,z>0$) with $xy-1,xz-1,yz-1$ all prime and $x$ among $2,6,10$.

• _Zhi-Wei Sun_

conjecture_5b. Conjecture (iii), part 2: Any integer $n>2$ not equal to $16$ can be written as $x+y+z$ ($x,y,z>0$) with $xy+1,xz+1,yz+1$ all prime and $x$ among $1,2,6$.

• _Zhi-Wei Sun_

Number of ways to write $n=x+y+z$ with $0<x\le y\le z$ such that all the six numbers $2x-1,2y-1,2z-1,2xy-1,2xz-1,2yz-1$ are prime.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.OEIS.«229969» (7 statements).

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

theorem conjecture_2 (n : ℕ) (hn : n > 6) :
    ∃ x y z : ℕ, 0 < y ∧ 0 < z ∧ x + y + z = n ∧
      x ∈ ({3, 4, 6, 10, 15} : Finset ℕ) ∧
      (2 * y - 1).Prime ∧ (2 * z - 1).Prime ∧
      (2 * x * y - 1).Prime ∧ (2 * x * z - 1).Prime ∧ (2 * y * z - 1).Prime

theorem conjecture_3a (n : ℕ) (hn : n > 6) :
    ∃ x y z : ℕ, 0 < x ∧ 0 < y ∧ 0 < z ∧ x + y + z = n ∧
      x ∈ ({2, 3, 4} : Finset ℕ) ∧
      (2 * x - 1).Prime ∧ (2 * y - 1).Prime ∧ (2 * z - 1).Prime ∧
      (2 * x * y * z - 1).Prime

theorem conjecture_3b (n : ℕ) (hn : n > 2) :
    ∃ x y z : ℕ, 0 < x ∧ 0 < y ∧ 0 < z ∧ x + y + z = n ∧
      x ∈ ({1, 2, 3} : Finset ℕ) ∧
      (2 * x + 1).Prime ∧ (2 * y + 1).Prime ∧ (2 * z + 1).Prime ∧
      (2 * x * y * z + 1).Prime

theorem conjecture_4 (n : ℕ) (hn : n > 4) :
    ∃ x y z : ℕ, 0 < y ∧ 0 < z ∧ x + y + z = n ∧
      (x = 3 ∨ x = 6) ∧
      (2 * y + 1).Prime ∧ (2 * x * y * z - 1).Prime ∧ (2 * x * y * z + 1).Prime

theorem conjecture_5a (n : ℕ) (hn : n > 5) :
    ∃ x y z : ℕ, 0 < x ∧ 0 < y ∧ 0 < z ∧ x + y + z = n ∧
      x ∈ ({2, 6, 10} : Finset ℕ) ∧
      (x * y - 1).Prime ∧ (x * z - 1).Prime ∧ (y * z - 1).Prime

theorem conjecture_5b (n : ℕ) (hn : n > 2) (hne : n ≠ 16) :
    ∃ x y z : ℕ, 0 < x ∧ 0 < y ∧ 0 < z ∧ x + y + z = n ∧
      x ∈ ({1, 2, 6} : Finset ℕ) ∧
      (x * y + 1).Prime ∧ (x * z + 1).Prime ∧ (y * z + 1).Prime

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

• A229969

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.