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Four-square conjecture with powers of 2, 3, and 5

Any integer $n > 1$ can be written as $(2^a \cdot 3^b)^2 + (2^c \cdot 5^d)^2 + x^2 + y^2$ where $a, b, c, d, x, y$ are nonnegative integers.

Zhi-Wei Sun has offered a $2,500 prize for the first proof.

References:

    A308734

    Z.-W. Sun, "Refining Lagrange's four-square theorem," J. Number Theory 175 (2017), 167-190. https://doi.org/10.1016/j.jnt.2016.11.008

    Z.-W. Sun, "Restricted sums of four squares," Int. J. Number Theory 15 (2019), 1863-1893.

    Z.-W. Sun, "Various Refinements of Lagrange's Four-Square Theorem," Westlake Number Theory Symposium, Nanjing University, China, 2020.

    S. Banerjee, "On a conjecture of Sun about sums of restricted squares," J. Number Theory 256 (2024), 253-289.

namespace OeisA308734

The predicate that n can be written as $(2^a \cdot 3^b)^2 + (2^c \cdot 5^d)^2 + x^2 + y^2$ for nonnegative integers $a, b, c, d, x, y$.

def A (n : ℕ) : Prop := ∃ a b c d x y : ℕ, n = (2 ^ a * 3 ^ b) ^ 2 + (2 ^ c * 5 ^ d) ^ 2 + x ^ 2 + y ^ 2 @[category test, AMS 11] theorem a_2 : A 2 := ⟨0, 0, 0, 0, 0, 0, ⊢ 2 = (2 ^ 0 * 3 ^ 0) ^ 2 + (2 ^ 0 * 5 ^ 0) ^ 2 + 0 ^ 2 + 0 ^ 2 All goals completed! 🐙⟩ @[category test, AMS 11] theorem a_3 : A 3 := ⟨0, 0, 0, 0, 0, 1, ⊢ 3 = (2 ^ 0 * 3 ^ 0) ^ 2 + (2 ^ 0 * 5 ^ 0) ^ 2 + 0 ^ 2 + 1 ^ 2 All goals completed! 🐙⟩ @[category test, AMS 11] theorem a_4 : A 4 := ⟨0, 0, 0, 0, 1, 1, ⊢ 4 = (2 ^ 0 * 3 ^ 0) ^ 2 + (2 ^ 0 * 5 ^ 0) ^ 2 + 1 ^ 2 + 1 ^ 2 All goals completed! 🐙⟩ @[category test, AMS 11] theorem a_5 : A 5 := ⟨1, 0, 0, 0, 0, 0, ⊢ 5 = (2 ^ 1 * 3 ^ 0) ^ 2 + (2 ^ 0 * 5 ^ 0) ^ 2 + 0 ^ 2 + 0 ^ 2 All goals completed! 🐙⟩ @[category test, AMS 11] theorem a_6 : A 6 := ⟨0, 0, 0, 0, 2, 0, ⊢ 6 = (2 ^ 0 * 3 ^ 0) ^ 2 + (2 ^ 0 * 5 ^ 0) ^ 2 + 2 ^ 2 + 0 ^ 2 All goals completed! 🐙⟩

Zhi-Wei Sun's Four-Square Conjecture (A308734): Any integer $n > 1$ can be written as $(2^a \cdot 3^b)^2 + (2^c \cdot 5^d)^2 + x^2 + y^2$ for nonnegative integers $a, b, c, d, x, y$.

@[category research open, AMS 11] theorem declaration uses 'sorry'conjecture (n : ℕ) (hn : 1 < n) : A n := n:ℕhn:1 < n⊢ A n All goals completed! 🐙 end OeisA308734