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import FormalConjecturesUtilSum of a triangular number, a generalized pentagonal number, and a generalized heptagonal number
Any nonnegative integer can be written as $x(x+1)/2 + y(3y+1)/2 + z(5z+1)/2$ with $x, y, z$ nonnegative integers.
Zhi-Wei Sun has offered a USD 135 prize for the first proof of this conjecture.
Zhi-Wei Sun, "Universal sums of three quadratic polynomials", arXiv:1502.03056 [math.NT]
namespace OeisA287616
The predicate that n can be written as $x(x+1)/2 + y(3y+1)/2 + z(5z+1)/2$ for
nonnegative integers $x, y, z$.
def A (n : ℕ) : Prop :=
∃ x y z : ℕ, n = x * (x + 1) / 2 + y * (3 * y + 1) / 2 + z * (5 * z + 1) / 2
@[category test, AMS 11]
theorem a_0 : A 0 :=
⟨0, 0, 0, ⊢ 0 = 0 * (0 + 1) / 2 + 0 * (3 * 0 + 1) / 2 + 0 * (5 * 0 + 1) / 2 All goals completed! 🐙⟩
@[category test, AMS 11]
theorem a_1 : A 1 :=
⟨1, 0, 0, ⊢ 1 = 1 * (1 + 1) / 2 + 0 * (3 * 0 + 1) / 2 + 0 * (5 * 0 + 1) / 2 All goals completed! 🐙⟩
@[category test, AMS 11]
theorem a_2 : A 2 :=
⟨0, 1, 0, ⊢ 2 = 0 * (0 + 1) / 2 + 1 * (3 * 1 + 1) / 2 + 0 * (5 * 0 + 1) / 2 All goals completed! 🐙⟩
@[category test, AMS 11]
theorem a_3 : A 3 :=
⟨2, 0, 0, ⊢ 3 = 2 * (2 + 1) / 2 + 0 * (3 * 0 + 1) / 2 + 0 * (5 * 0 + 1) / 2 All goals completed! 🐙⟩
@[category test, AMS 11]
theorem a_4 : A 4 :=
⟨1, 0, 1, ⊢ 4 = 1 * (1 + 1) / 2 + 0 * (3 * 0 + 1) / 2 + 1 * (5 * 1 + 1) / 2 All goals completed! 🐙⟩
Zhi-Wei Sun's Conjecture (A287616): Any nonnegative integer can be written as the sum of a triangular number $x(x+1)/2$, a generalized pentagonal number $y(3y+1)/2$, and a generalized heptagonal number $z(5z+1)/2$, where $x, y, z$ are nonnegative integers.
@[category research open, AMS 11]
theorem conjecture (n : ℕ) : A n := n:ℕ⊢ A n
All goals completed! 🐙
end OeisA287616