A rather beautiful conjecture
Discovered (almost) entirely by accident

By Tom Boardman and Jamie Sawyer
With thanks to: Andrew Ewert

The Story

Early in the Spring Term of the 2007/08 academic year, the Warwick Mathematics Society (WMS) was running one of its weekly "Discussion Groups" events, wherein an undergrad or postgrad mathematician would introduce an interesting and complicated area of maths to a group of fellow undergrad students. As was traditional, the group left to go to the on-campus pub at the end of the evening to discuss some of the ideas in a more casual setting.

During this evening, Andrew Ewert - a first year undergrad student, who had gained the nickname "PBIG" or "Proof-By-Induction Guy" for his numerous incorrect proofs of major conjectures by induction - came to the group with an idea for proving the Goldbach Conjecture by induction. After taking a while to work through his slightly confused idea, we established what he thought the trivial statement at the base of his proof was:

Clearly this was not a trivial statement, and in fact many of us believed it to be untrue but, due to our ever more inebriated state, we could not come up with a counterexample at the time. Thinking about the conjecture was then put on hold until the next day when we could attack it with clear heads!

The following day, Sam Derbyshire put a post up on the WMS forum with the results of a small python script he had produced to find any counterexamples up to 1000 (including 4 as a test variable) - this was the output and Sam's explanitory post:

[4, 38, 68, 80, 98, 122, 128, 146, 158, 164, 188, 206, 212, 218, 224, 248, 278, 290, 302, 308, 326, 332, 338, 344, 368, 374, 380, 398, 410, 416, 428, 440, 458, 476, 488, 500, 518, 530, 536, 542, 548, 554, 578, 584, 608, 614, 626, 632, 638, 668, 674, 692, 698, 710, 716, 728, 734, 740, 752, 758, 770, 782, 788, 794, 806, 818, 836, 848, 854, 872, 878, 896, 902, 908, 920, 926, 938, 962, 968, 992, 998]
For example, the only prime decompositions of 38 are :
38 = 31 + 7
33 and 9 are both composite
38 = 19 + 19
21 is composite

Ah well, another conjecture debunked - time to move on with our lives! ...

...or was it?

For the next few days silly additional assumptions were added (see the forum page linked to at the bottom), but nothing came to fruition. 12 days after the original conjecture was forumalted, the WMS Academic Support Officer, Tom Boardman, was browsing the internet while somewhat tiddly and stumbled across the thread once more. To quote his post in its entirity:

I would like to conjecture that every counterexample to pBig's conjecture is of the form 38+6n

A counterexample was assumed certain - after all, this was yet another silly additional assumtion, powered by alcohol no less!

The first surprise came in 5 minutes - there were no counterexamples below 10,000 to this new conjecture. Sam's python script then tackled bigger numbers - 30,000; 100,000 - but these both fell soon after. The script was then run for 8 hours to attempt to find a counterexample under 1,000,000 - to the surprise of most, this did not appear.

After a consultation with the resident number theorist at Warwick University, Samir Siksek, it was decided that we needed a bigger boat.

Jamie Sawyer and Tom Boardman then set to work on a new algorithm, this time running on C rather than the (slightly questionable, computationally speaking) python language. After some hiccups with naive use of arrays, a linked-list version of the algorithm hit the 10 million mark in about an hour on Jamie's desktop computer - much improvement. Celebration ensued, but the team did not rest on their laurels - over the next few days 50 million was downed (in just 38 hours). However, bad news was just around the corner...

Tom and Jamie went to bring this achievement to Samir Siksek and ask where we should go next. It was at this point that they were told that we should be aiming for around 1,000,000,000 to even think about announcing the result. Given that hitting just 50,000,000 took us 38 hours, this new obstacle seemed insurmountable.

Much programming, many errors and around 3 days total debugging time later, we had improved the algorithm massively - 10,000,000 took 25 seconds, 100,000,000 took just 9 minutes, and even 1,000,000,000 was taken down in just 3 hours 15 minutes.

As currently stands, under the limitation of an (albeit powerful) desktop computer, we have proven the Boardman-Ewert Conjecture for numbers below 10,000,000,000.