1. FormalConjectures
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  3. «303656»
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Sum of two squares, a power of 3, and a power of 5

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

Zhi-Wei Sun has offered a $3,500 prize for the first proof.

References:

    A303656

    Z.-W. Sun, "Restricted sums of four squares," arXiv preprint: https://arxiv.org/abs/1701.05868v10

    Z.-W. Sun, "Refining Lagrange's four-square theorem," Journal of Number Theory: http://maths.nju.edu.cn/~zwsun/RefineFourSquareTh.pdf

    Z.-W. Sun, "Restricted sums of three or four squares": http://maths.nju.edu.cn/~zwsun/Square-sum.pdf

    Zhi-Wei Sun's 1-3-5 conjecture and variations: https://www.aimspress.com/aimspress-data/era/2020/2/PDF/1935-9179_2020_2_589.pdf

namespace OeisA303656

The predicate that n can be written as $a^2 + b^2 + 3^c + 5^d$ for nonnegative integers.

def A (n : ℕ) : Prop := ∃ a b c d : ℕ, n = a ^ 2 + b ^ 2 + 3 ^ c + 5 ^ d @[category test, AMS 11] theorem a_2 : A 2 := ⟨0, 0, 0, 0, ⊢ 2 = 0 ^ 2 + 0 ^ 2 + 3 ^ 0 + 5 ^ 0 All goals completed! 🐙⟩ @[category test, AMS 11] theorem a_3 : A 3 := ⟨1, 0, 0, 0, ⊢ 3 = 1 ^ 2 + 0 ^ 2 + 3 ^ 0 + 5 ^ 0 All goals completed! 🐙⟩ @[category test, AMS 11] theorem a_4 : A 4 := ⟨1, 1, 0, 0, ⊢ 4 = 1 ^ 2 + 1 ^ 2 + 3 ^ 0 + 5 ^ 0 All goals completed! 🐙⟩ @[category test, AMS 11] theorem a_5 : A 5 := ⟨0, 1, 1, 0, ⊢ 5 = 0 ^ 2 + 1 ^ 2 + 3 ^ 1 + 5 ^ 0 All goals completed! 🐙⟩ @[category test, AMS 11] theorem a_6 : A 6 := ⟨0, 0, 0, 1, ⊢ 6 = 0 ^ 2 + 0 ^ 2 + 3 ^ 0 + 5 ^ 1 All goals completed! 🐙⟩ @[category test, AMS 11] theorem a_25 : A 25 := ⟨1, 4, 1, 1, ⊢ 25 = 1 ^ 2 + 4 ^ 2 + 3 ^ 1 + 5 ^ 1 All goals completed! 🐙⟩

Zhi-Wei Sun's Conjecture (A303656): Any integer $n > 1$ can be written as the sum of two squares, a power of 3, and a power of 5.

@[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 OeisA303656