Along with Pomerance and Wagstaff, he is the originator of what has come to be known as "the $620 problem", the question of whether there is a number which is simultaneously a strong pseudoprime to the base 2 and a Lucas pseudoprime. If the answer is yes, he promised to pay $500 for the solution, with Wagstaff paying $100 and Pomerance $20. If the answer is confirmed as no, Selfridge would be on the hook for $20, Wagstaff $100 and Pomerance $500.
When asked why he would volunteer higher amount upon production of a counterexample -- which almost surely exists -- he explained that if someone produced a dense proof claiming to show that none existed, he would rather Pomerance -- with $500 on the line -- have to read through the proof to find an error. The counterexample, on the other hand, would be easy to verify or dismiss.
Throughout the '90s, and even into this century as his health was failing, I ran into Selfridge at practically every number theory conference I attended. Having retired, he enjoyed nothing better than traveling around, listening to mathematics talks, and enjoying the social company of mathematicians (with the associated libations). Having invested well, he decided he would rather the money go to mathematics than to the government through taxes, so he set up the Number Theory Foundation, which helps fund many of the conferences he enjoyed.
Rest in Peace, John Selfridge. You will be missed.
