A solution to the problem was given by Richard Stong in Vol. 128, November 2021 issue p. 858, namely d+1 is a square.
When d=m2−1,m>2, there are usually 4 or 8 Nagell equivalence classes of solutions. However when m=33539, there are at least 10 Nagell equivalence classes of solutions.
We always have the four solution classes given by (±m,1) and (±(m3−m2−m),m−1).
We list the m in the range 2<m≤3000 which give 8 solution classes, listing only the extra 4 classes for each such m.
With the help of the OEIS, when m=2n2+2n−1,n≥2, we found the following pairs of fundamental solutions:
(±m(2n3+2n2−2n−1,n),(±m(2n3+4n2−1),n+1)
which we call Type 1 fundamental solutions.
We call the remaining solution pairs in the table, Type 2 fundamental solutions.
Here are some Type 2 fundamental solutions:
(i) (m(4n4+4n3−5n2−3n+1),n) and (m(8n5+16n4−2n3−12n2+1),2n2+2n−1),m=4n3+4n2−3n−1
(ii) (m(4n4+12n3+7n2−3n−1),n+1) and (m(8n5+24n4+14n3−10n2−6n+1),2n2+2n−1),m=4n3+8n2+n−2.
Solutions (i) and (ii) were discovered after noticing that with d=m2−1, the equation x2−(d2+d)y2=1−d2 implies m divides x.
Then with x=mX, we get the simpler equation X2−(m2−1)y2=2−m2, which was studied by Kenji Kashihara in 1990 and 1994. (See references.)
The equation can also be written as X2−1=(m2−1)(y2−1).
This diophantine equation usually has 2 or 4 fundamental solutions, but has 6 when m=33539, namely (±1,1),(±4326401,129),(±669941,20).
The Type 2 solutions in the table below with m=153 and m=373 are not covered by the previous formulae.
n=2m=11(±209,2)(±341,3)n=3m=23(±1495,3)(±2047,4)n=4m=39(±5889,4)(±7449,5)Type 2m=41(±2911,2)(±18409,11)n=5m=59(±17051,5)(±20591,6)Type 2m=64(±11584,3)(±44864,11)n=6m=83(±40753,6)(±47725,7)n=7m=111(±85359,7)(±97791,8)Type 2m=134(±50786,3)(±412586,23)n=8m=143(±162305,8)(±182897,9)Type 2m=153(±40545,2)(±959463,41)n=9m=179(±286579,9)(±318799,10)Type 2m=181(±126881,4)(±752779,23)n=10m=219(±477201,10)(±525381,11)n=11m=263(±757703,11)(±827135,12)Type 2m=307(±365023,4)(±3674483,39)n=12m=311(±1156609,12)(±1253641,13)n=13m=363(±1707915,13)(±1840047,14)Type 2m=373(±393515,3)(±8903137,64)Type 2m=386(±729926,5)(±5808914,39)n=14m=419(±2451569,14)(±2627549,15)n=15m=479(±3433951,15)(±3663871,16)n=16m=543(±4608353,16)(±5003745,17)Type 2m=571(±564719,2)(±49883131,153)Type 2m=584(±1670824,5)(±20119384,59)n=17m=611(±56335459,17)(±6709391,18)n=18m=683(±8383825,18)(±8850997,19)Type 2m=703(±2923777,6)(±29154113,59)n=19m=759(±10930359,,19)(±11507199,20)Type 2m=781(±1725229,3)(±81732431,134)n=20m=839(±14060801,20)(±1476556,21)Type 2m=900(±8873100,11)(±33200100,41)n=21m=923(±17870203,21)(±18723055,22)Type 2m=989(±5786639,6)(±81178109,83)n=22m=1011(±22463409,22)(±23486541,23)n=23m=1103(±27955535,23)(±29173247,24)Type 2m=1156(±9258404,7)(±110907796,83)n=24m=1199(±34472449,24)(±35911249,25)n=25m=1299(±42151251,25)(±43839951,26)n=26m=1403(±51140753,26)(±53110565,27)Type 2m=1405(±21624355,11)(±126322145,64)Type 2m=1425(±7864575,4)(±367537425,181)n=27m=1511(±61601959,27)(±63886591,28)Type 2m=1546(±16559206,7)(±265292054,111)n=28m=1623(±73708545,28)(±76344297,29)n=29m=1739(±87647339,29)(±90673199,30)Type 2m=1769(±24838529,8)(±347344919,111)n=30m=1859(±103618801,30)(±107076541,31)n=31m=1983(±121837503,31)(±125771775,32)n=32m=2111(±142532609,32)(±146991041,33)Type 2m=2131(±7865521,2)(±2592998669,571)Type 2m=2174(±13367926,3)(±1762894426,373)n=33m=2243(±165948355,33)(±170981647,34)Type 2m=2279(±41224831,8)(±742701031,143)n=34m=2379(±192344529,34)(±198006549,35)Type 2m=2417(±22625537,4)(±1793450255,307)n=35m=2519(±221996951,35)(±228344831,36)Type 2m=2566(±58892266,9)(±941539814,143)n=36m=2663(±255197953,36)(±262292185,37)n=37m=2811(±292256859,37)(±300161391,38)n=38m=2963(±333500465,38)(±342282797,39)
References
Last modified 29th September 2023