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fix(Wikipedia/WallSunSun): restrict D to fundamental discriminants #1813

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fix(Wikipedia/WallSunSun): restrict D to fundamental discriminants #1813

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fix(Wikipedia/WallSunSun): restrict `D` to fundamental discriminants
YaelDillies Jan 21, 2026
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fix
YaelDillies Jan 21, 2026
f1d8b64
exclude 1
YaelDillies Jan 21, 2026
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YaelDillies Jan 26, 2026
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12 changes: 7 additions & 5 deletions FormalConjectures/Wikipedia/WallSunSun.lean
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Expand Up @@ -41,14 +41,16 @@ theorem infinite_isWallSunSunPrime : {p : ℕ | IsWallSunSunPrime p}.Infinite :=
sorry

/--
A Lucas–Wieferich prime associated with $(a,b)$ is an odd prime $p$, not dividing $a^2 - b$, such
A Lucas–Wieferich prime associated with $(a,b)$ is an odd prime $p$, not dividing $a^2 - 4b$, such
that $U_{p-\varepsilon}(a,b) \equiv 0 \pmod{p^2}$ where $U(a,b)$ is the Lucas sequence of the first
kind and $\varepsilon$ is the Legendre symbol $\left({\tfrac {a^2-4b}{p}}\right)$.
The discriminant of this number is the quantity $a^2 - 4b$. It is conjectured that there are
infinitely many Lucas–Wieferich primes of any given discriminant.
infinitely many Lucas–Wieferich primes of any given non-one fundamental discriminant.

TODO: Source this conjecture
-/
@[category research open, AMS 11]
theorem infinite_isWallSunSunPrime_of_disc_eq {D : ℕ} (hD₀ : 0 < D)
(hD : D ≡ 0 [MOD 4] ∨ D ≡ 1 [MOD 4]) :
{p : ℕ | ∃ a b : ℕ, a ^ 2 - 4 * b = D ∧ IsLucasWieferichPrime a b p}.Infinite := by
theorem infinite_isWallSunSunPrime_of_disc_eq {D : ℤ} (hD : IsFundamentalDiscr D)
(hD₁ : D ≠ 1) :
{p : ℕ | ∃ a b, a ^ 2 - 4 * b = D ∧ IsLucasWieferichPrime a b p}.Infinite := by
sorry
16 changes: 1 addition & 15 deletions FormalConjecturesForMathlib.lean
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@@ -1,19 +1,4 @@
/-
Copyright 2026 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 FormalConjecturesForMathlib.Algebra.GCDMonoid.Finset

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import FormalConjecturesForMathlib.Algebra.Group.Action.Pointwise.Set.Basic
import FormalConjecturesForMathlib.Algebra.Group.Indicator
import FormalConjecturesForMathlib.Algebra.Order.Group.Pointwise.Interval
Expand Down Expand Up @@ -70,6 +55,7 @@
import FormalConjecturesForMathlib.NumberTheory.AdditivelyComplete
import FormalConjecturesForMathlib.NumberTheory.CoveringSystem
import FormalConjecturesForMathlib.NumberTheory.DirichletCharacter.Basic
import FormalConjecturesForMathlib.NumberTheory.FundamentalDiscr
import FormalConjecturesForMathlib.NumberTheory.Lacunary
import FormalConjecturesForMathlib.NumberTheory.LegendreSymbol.Basic
import FormalConjecturesForMathlib.NumberTheory.PrimeGap
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