Not off the top of my head, but since a proof of Collatz does not require working under these constraints, I don’t think the distinction has any important implications.
Actually, Collatz DOES require working under these constraints if staying in arithmetic. The conjecture itself needs universal quantification (“for ALL numbers...”) to even state it. In pure arithmetic we can only verify specific cases: “4 goes to 2″, “5 goes to 16 to 8 to 4 to 2”, etc.
Alright, so Collatz will be proved, and the proof will not be done by “staying in arithmetic”. Just as the proof that there do not exist numbers p and q that satisfy the equation p² = 2 q² (or equivalently, that all numbers do not satisfy it) is not done by “staying in arithmetic”. It doesn’t matter.
This shows why it DOES matter: Both Collatz and p² = 2q² require going above arithmetic to even state them (“for ALL numbers...”). That’s the key insight—we can’t even formulate these interesting conjectures within pure arithmetic, let alone prove them. In pure arithmetic we can only check specific cases:
Actually, your example still goes beyond arithmetic:
“No integer solutions” is a universal statement about ALL integers
Proof by contradiction is still needed
Even reframed, it requires proving properties about ALL possible p and q
In pure arithmetic we can only check specific cases: “3² ≠ 2×2²”, “4² ≠ 2×3²”, etc. Any examples using just counting and basic operations?
Not off the top of my head, but since a proof of Collatz does not require working under these constraints, I don’t think the distinction has any important implications.
Actually, Collatz DOES require working under these constraints if staying in arithmetic. The conjecture itself needs universal quantification (“for ALL numbers...”) to even state it. In pure arithmetic we can only verify specific cases: “4 goes to 2″, “5 goes to 16 to 8 to 4 to 2”, etc.
Alright, so Collatz will be proved, and the proof will not be done by “staying in arithmetic”. Just as the proof that there do not exist numbers p and q that satisfy the equation p² = 2 q² (or equivalently, that all numbers do not satisfy it) is not done by “staying in arithmetic”. It doesn’t matter.
This shows why it DOES matter: Both Collatz and p² = 2q² require going above arithmetic to even state them (“for ALL numbers...”). That’s the key insight—we can’t even formulate these interesting conjectures within pure arithmetic, let alone prove them. In pure arithmetic we can only check specific cases:
“4 goes to 2”
“3² ≠ 2×2²”