Conjectures Disproven by the use of Computers?
Question: Is there a list of conjectures (famous or not so famous) that were shown to be false by employing the use of computers?
This is just curiosity more than anything. I was actually wondering if more often than not - computers show many conjectures to be false? This question should include
- The conjectured existence of mathematical structures, for example in finite geometry
- Any instance of a computer (no matter the language) handling "intricate calculations" that would otherwise take to long or be impossible to do by hand. This for example would cover all instances where there is a theoretical set-up and a final result established by a computer calculation. It would also cover refutations where some results, not all of them, required a computer.
- A computer "showed" the conjecture was false, via something like AI ?
- The counterexample does not have to be large. An example of this would be something along the lines - "the conjecture is true for the first 3 integers" but a computer showed it is false for the fourth one.
I started an initial list by GOOGLING and try to organize by broad categories.
Groups, Graphs and Geometry
- 1. There is No McLaughlin Geometry
- 2. A counterexample to the pseudo 2-factor isomorphic graph conjecture
- 3. The $0-1$ Conjecture is false
- 4. A Counterexample to the Hirsch Cnojecture
- 5. COUNTEREXAMPLES TO THE POSET CONJECTURES OF NEGGERS, STANLEY, AND STEMBRIDGE
- 6. Counterexample To Wall's Conjecture
- 7. A Counterexample To Tait's Hamiltonian Graph Conjecture
Number Theory
- 1. A Disproof of Polya's Conjecture
- 2. Disproof of the Merten's Conjecture
- 3. A Counterexample to Euler's Sum of Powers Conjecture
- 4. RADEMACHER’S INFINITE PARTIAL FRACTION CONJECTURE IS (almost certainly) FALSE - Doron
Analysis
Solution 1:
Until recently, it seemed that whenever two natural numbers are amicable, then they have the same smallest prime factor. If this was true, then it would follow that:
- there are no coprime amicable numbers;
- there is no example of two amicable numbers such that one of them is even and the other one is odd.
However, in October 2015 a computer found an amicable pair:$$(445\,953\,248\,528\,881\,275,659\,008\,669\,204\,392\,325)$$such that the smallest prime factor of the first term is $3$, whereas the smallest prime factor of the second term is $5$.