feat: add Sophie Germain theorem formalization

[kellyzhao1231]

A Lean 4 + Mathlib formalization of Sophie Germain's classical theorem on Fermat's Last Theorem, translated from the Isabelle/HOL formalization by Benoît Ballenghien (Université Paris-Saclay, CNRS, ENS Paris-Saclay, LMF, 2026).

[github-actions]

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[github-actions]

PR summary 044631201f

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference
Mathlib.NumberTheory.SophieGermain.SophieGermain (new file)	1

---

Declarations diff

+ Int.ModEq.finset_sum
+ Int.gcd_div_gcd_eq_one
+ Nat.eq_pow_of_coprime_mul_pow
+ SophieGermainPrime
+ SophieGermainSum
+ dvd_natAbs_of_natCast_dvd
+ eq_pow_or_neg_of_coprime_mul_pow
+ ge_three
+ int_dvd_gcd
+ int_gcd_dvd_left
+ int_gcd_dvd_right
+ natCast_dvd_of_dvd_natAbs
+ ne_zero
+ not_dvd_of_not_dvd_div
+ odd
+ p_not_congr_neg_one
+ p_not_congr_one
+ p_not_congr_zero
+ pairwise_coprime_of_fermat_eq
+ pos
+ pow_congr_one_or_neg_one
+ prime
+ prime_twop_succ
+ q_dvd_of_fermat_triple
+ sgSum_congr_mod
+ sgSum_mul_eq_pow_add_pow
+ sophie_germain_factorization
+ sophie_germain_theorem
+ sum_dvd_sgSum_sub_scalar

You can run this locally as follows

## summary with just the declaration names:
./scripts/pr_summary/declarations_diff.sh <optional_commit>

## more verbose report:
./scripts/pr_summary/declarations_diff.sh long <optional_commit>

The doc-module for scripts/pr_summary/declarations_diff.sh contains some details about this script.

---

No changes to technical debt.

You can run this locally as

./scripts/reporting/technical-debt-metrics.sh pr_summary

• The relative value is the weighted sum of the differences with weight given by the inverse of the current value of the statistic.
• The absolute value is the relative value divided by the total sum of the inverses of the current values (i.e. the weighted average of the differences).

[grunweg]

Hello from the triage team! Per the AI guidelines linked above, can you clarify if/to what extent you used AI to write this code? (Also, you probably need to run lake exe mk_all to make sure your new file is picked up by Lean.) Thanks!