Let $C$ be a Carmichael-number and $D$ the next larger Carmichael-number. Define $$d:=D-C$$ the "Carmichael-gap" analogue to the prime-gap and define $$r(C):=\frac{\ln(d)}{ln(C)}$$
Is this measure $r$ for the magnitude of the Carmichael-gaps bounded from below by some positive constant ? Or can it get arbitary small ?
The "jumping champions" upto $10^{16}$ are :
561 1105 544 0.99513854415920529069297505122613733452
1105 1729 624 0.91845279474907703929405663875235214559
1729 2465 736 0.88544142550502624476809166095976954681
2465 2821 356 0.75223694422010723491272040202550791362
62745 63973 1228 0.64390774746904012158853991518318110372
656601 658801 2200 0.57456582883692295131879220385500549080
4903921 4909177 5256 0.55610659670066835361057221495440590042
6049681 6054985 5304 0.54921121250212586428868981623071540180
8719309 8719921 612 0.40152130566050285575474067969645202654
214850881 214852609 1728 0.38856102392325088026485935143416117813
448238792737 448238823901 31164 0.38567133459503335323705627312210629442
4694439323701 4694439367801 44100 0.36652392379576449406917523790330483820
593961500246401 593961500298673 52272 0.31936823540412766301267893738903335990
I remember a gap from a user that probably beats this below $2^{64}$.
How can we find small gaps systematically ? Or do we just have to go through all the Carmichael-numbers to find the most spectacular examples ?