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High school

A hypothetical counterexample

Assume a forbidden solution exists, then keep its evidence together.

Start with a concrete example

“Every bird can fly” would be disproved by one bird that cannot. In mathematics too, one genuine example that satisfies the conditions but breaks the conclusion disproves an “every” claim.

A small check: try the idea

Think with the concrete example above. You can try again.

To disprove “every allowed object has property P”, what must an example do?

Read the answer and reasoning

Meet the conditions and fail P. A bird that cannot fly matters to “all birds fly” because it is still a bird. Breaking the hypotheses does not give a counterexample.

Build the idea in three steps

  1. A counterexample would be one permitted input for which a claimed universal statement fails.
  2. We assume such an input exists temporarily. This is not a discovered solution to FLT.
  3. A Frey package records normalized integer data, a prime exponent, the forbidden equality, and extra conditions needed by later arguments.

A small example

If a hypothetical solution created a package, and no such package can exist, that solution is impossible too.

Lean definition · original excerpt

FLT · aa2d8b3 · Def_FLTPrelim_FreyPackage.lean

structure FreyPackage where   a : ℤ   b : ℤ   c : ℤ  ha0 : a ≠ 0  hb0 : b ≠ 0  hc0 : c ≠ 0   p : ℕ  pp : Nat.Prime p  hp5 : 5 ≤ p   hFLT : a ^ p + b ^ p = c ^ p   hgcdab : gcd a b = 1   ha4 : (a : ZMod 4) = 3   hb2 : (b : ZMod 2) = 0

How to read this Lean

hFLT records the assumed equation. pp, hp5, hgcdab, ha4 and hb2 are different required conditions, not conclusions silently assumed to be true.

Original source ↗ · L17–39Open the complete saved source

Where this idea is used

1. The impossible order2. The case of three3. Too many suspects4. The suitcase diet5. The suspicious parcel6. The scissors incident7. The veterans’ cabinet8. The secret duet9. Two doors to the concert10. A three-note audition11. Borrow a better passport12. The workshops agree13. Pip’s impossible quilt14. The shrinking reservation16. The room with no room
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A hypothetical counterexample

One thing at a time

Read the three explanation steps first. The Lean source is there for a closer look afterwards; you do not have to read it first.

In simpler words

Assume a forbidden solution exists, then keep its evidence together.

Earlier ideas are listed under “Start with these ideas” on this page.

A small clue

If a hypothetical solution created a package, and no such package can exist, that solution is impossible too.

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