FormalConjectures.ErdosProblems.«324»

/-
Copyright 2025 The Formal Conjectures Authors.

Licensed under the Apache License, Version 2.0 (the "License");
you may not use this file except in compliance with the License.
You may obtain a copy of the License at

    https://www.apache.org/licenses/LICENSE-2.0

Unless required by applicable law or agreed to in writing, software
distributed under the License is distributed on an "AS IS" BASIS,
WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
See the License for the specific language governing permissions and
limitations under the License.
-/

import FormalConjecturesUtil

Erdős Problem 324

Reference: erdosproblems.com/324


open scoped Polynomial


namespace Erdos324

Does there exist a polynomial $f(x)\in\mathbb{Z}[x]$ such that all the sums $f(a)+f(b)$ with $a < b$ nonnegative integers are distinct?

@[category research open, AMS 11]
theorem declaration uses 'sorry'erdos_324 : answer(sorry) ↔
    ∃ f : ℤ[X], {(a, b) : ℕ × ℕ | a < b}.InjOn fun (a, b) => f.eval (a : ℤ) + f.eval (b : ℤ) := by⊢ True ↔
  ∃ f,
    Set.InjOn
      (fun x =>
        match x with
        | (a, b) => Polynomial.eval (↑a) f + Polynomial.eval (↑b) f)
      {(a, b) | a < b}
  sorryAll goals completed! 🐙

Probably $f(x) = x^5$ has the property that the sums $f(a)+f(b)$ with $a < b$ nonnegative integers are distinct.

@[category research open, AMS 11]
theorem declaration uses 'sorry'erdos_324.variants.quintic : {(a, b) : ℕ × ℕ | a < b}.InjOn fun (a, b) => a ^ 5 + b ^ 5 := by⊢ Set.InjOn
  (fun x =>
    match x with
    | (a, b) => a ^ 5 + b ^ 5)
  {(a, b) | a < b}
  sorryAll goals completed! 🐙


end Erdos324