### A diophantine equation of Nikolai Osipov

Problem 12164 American Math. Monthly, February 2020, proposed by Nikolai Osipov, asked for a characterization of the positive integers $$d$$ such that the equation $$(d^2+d)x^2-y^2=d^2-1$$ has a solution in positive integers $$x$$ and $$y$$.

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 $$d=m^2-1, m>2$$, 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.

With $$d=m^2-1, m>2$$, we always have the four solution classes given by $$(\pm m,1)$$ and $$(\pm(m^3-m^2-m), m-1)$$.

We list the $$m$$ in the range $$2 < m \le 3000$$ which give $$8$$ solution classes, listing only the extra $$4$$ classes for each such $$m.$$

With the help of the OEIS, when $$m=2n^2+2n-1, n \geq 2,$$ we found the following pairs of fundamental solutions:

$$(\pm m(2n^3+2n^2-2n-1, n),\quad (\pm m(2n^3+4n^2-1), n+1)$$

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) $$(m(4n^4+4n^3-5n^2-3n+1), n)$$ and $$(m(8n^5+16n^4-2n^3-12n^2+1), 2n^2+2n-1), {\color{red}{m=4n^3+4n^2-3n-1}}$$

(ii) $$(m(4n^4+12n^3+7n^2-3n-1), n+1)$$ and $$(m(8n^5+24n^4+14n^3-10n^2-6n+1), 2n^2+2n-1), {\color{blue}{m=4n^3+8n^2+n-2}}.$$

Solutions (i) and (ii) were discovered after noticing that with $$d=m^2-1$$, the equation $$x^2-(d^2+d)y^2=1-d^2$$ implies $$m$$ divides $$x$$.
Then with $$x=mX$$, we get the simpler equation $$X^2-(m^2-1)y^2=2-m^2$$, which was studied by Kenji Kashihara in 1990 and 1994. (See references.)
The equation can also be written as $$X^2-1=(m^2-1)(y^2-1)$$.
This diophantine equation usually has 2 or 4 fundamental solutions, but has 6 when m=33539, namely $$(\pm1,1), (\pm4326401,129), (\pm669941,20)$$.

The Type 2 solutions in the table below with m=153 and m=373 are not covered by the previous formulae.

$\begin{array}{|c|c|r|r|}\hline n=2 & m=11 & (\pm 209,2) & (\pm 341,3)\\\hline n=3 & m=23 & (\pm 1495,3) & (\pm 2047,4)\\\hline n=4 & m=39 & (\pm 5889,4) & (\pm 7449,5)\\\hline Type\ 2&{\color{red} {m=41}} & (\pm 2911,2) & (\pm 18409,11)\\\hline n=5 & m=59 & (\pm 17051,5) & (\pm 20591,6)\\\hline Type\ 2& {\color{blue}{m=64}} & (\pm 11584,3) & (\pm 44864,11)\\\hline n=6 & m=83 & (\pm 40753,6) & (\pm 47725,7)\\\hline n=7 & m=111 & (\pm 85359,7) & (\pm 97791,8)\\\hline Type\ 2& {\color{red}{m=134}} & (\pm 50786,3) & (\pm 412586,23)\\\hline n=8 & m=143 & (\pm 162305,8) & (\pm 182897,9)\\\hline Type\ 2& m=153 & (\pm 40545,2) & (\pm 959463,41)\\\hline n=9 & m=179 & (\pm 286579,9) & (\pm 318799,10)\\\hline Type\ 2& {\color{blue}{m=181}} & (\pm 126881,4) & (\pm 752779,23)\\\hline n=10 & m=219 & (\pm 477201,10) & (\pm 525381,11)\\\hline n=11 & m=263 & (\pm 757703,11) & (\pm 827135,12)\\\hline Type\ 2& {\color{red}{m=307}} & (\pm 365023,4) & (\pm 3674483,39)\\\hline n=12 & m=311 & (\pm 1156609,12) & (\pm 1253641,13)\\\hline n=13 & m=363 & (\pm 1707915,13) & (\pm 1840047,14)\\\hline Type\ 2& m=373 & (\pm 393515,3) & (\pm 8903137,64)\\\hline Type\ 2& {\color{blue}{m=386}} & (\pm 729926,5) & (\pm 5808914,39)\\\hline n=14 & m=419 & (\pm 2451569,14) & (\pm 2627549,15)\\\hline n=15 & m=479 & (\pm 3433951,15) & (\pm 3663871,16)\\\hline n=16 & m=543 & (\pm 4608353,16) & (\pm 5003745,17)\\\hline Type\ 2& m=571 & (\pm 564719,2) & (\pm 49883131,153)\\\hline Type\ 2& {\color{red}{m=584}} & (\pm 1670824,5) & (\pm 20119384,59)\\\hline n=17 & m=611 & (\pm 56335459,17) & (\pm 6709391,18)\\\hline n=18 & m=683 & (\pm 8383825,18) & (\pm 8850997,19)\\\hline Type\ 2& {\color{blue}{m=703}} & (\pm 2923777,6) & (\pm 29154113,59)\\\hline n=19 & m=759 & (\pm 10930359,,19) & (\pm 11507199,20)\\\hline Type\ 2& m=781 & (\pm 1725229,3) & (\pm 81732431,134)\\\hline n=20 & m=839 & (\pm 14060801,20) & (\pm 1476556,21)\\\hline Type\ 2& m=900 & (\pm 8873100,11) & (\pm 33200100,41)\\\hline n=21 & m=923 & (\pm 17870203,21) & (\pm 18723055,22)\\\hline Type\ 2& {\color{red}{m=989}} & (\pm 5786639,6) & (\pm 81178109,83)\\\hline n=22 & m=1011 & (\pm 22463409,22) & (\pm 23486541,23)\\\hline n=23 & m=1103 & (\pm 27955535,23) & (\pm 29173247,24)\\\hline Type\ 2& {\color{blue}{m=1156}} & (\pm 9258404,7) & (\pm 110907796,83)\\\hline n=24 & m=1199 & (\pm 34472449,24) & (\pm 35911249,25)\\\hline n=25 & m=1299 & (\pm 42151251,25) & (\pm 43839951,26)\\\hline n=26 & m=1403 & (\pm 51140753,26) & (\pm 53110565,27)\\\hline Type\ 2& m=1405 & (\pm 21624355,11) & (\pm 126322145,64)\\\hline Type\ 2& m=1425 & (\pm 7864575,4) & (\pm 367537425,181)\\\hline n=27 & m=1511 & (\pm 61601959,27) & (\pm 63886591,28)\\\hline Type\ 2& {\color{red}{m=1546}} & (\pm 16559206,7) & (\pm 265292054,111)\\\hline n=28 & m=1623 & (\pm 73708545,28) & (\pm 76344297,29)\\\hline n=29 & m=1739 & (\pm 87647339,29) & (\pm 90673199,30)\\\hline Type\ 2& m=1769 & (\pm 24838529,8) & (\pm 347344919,111)\\\hline n=30 & m=1859 & (\pm 103618801,30) & (\pm 107076541,31)\\\hline n=31 & m=1983 & (\pm 121837503,31) & (\pm 125771775,32)\\\hline n=32& m=2111 & (\pm 142532609,32) & (\pm 146991041,33)\\\hline Type\ 2& m=2131 & (\pm 7865521,2) & (\pm 2592998669,571)\\\hline Type\ 2& m=2174 & (\pm 13367926,3) & (\pm 1762894426,373)\\\hline n=33& m=2243 & (\pm 165948355,33) & (\pm 170981647,34)\\\hline Type\ 2& {\color{red}{m=2279}} & (\pm 41224831,8) & (\pm 742701031,143)\\\hline n=34& m=2379 & (\pm 192344529,34) & (\pm 198006549,35)\\\hline Type\ 2& m=2417 & (\pm 22625537,4) & (\pm 1793450255,307)\\\hline n=35& m=2519 & (\pm 221996951,35) & (\pm 228344831,36)\\\hline Type\ 2& {\color{blue}{m=2566}} & (\pm 58892266,9) & (\pm 941539814,143)\\\hline n=36& m=2663 & (\pm 255197953,36) & (\pm 262292185,37)\\\hline n=37& m=2811 & (\pm 292256859,37) & (\pm 300161391,38)\\\hline n=38& m=2963 & (\pm 333500465,38) & (\pm 342282797,39)\\\hline \end{array}$ References

1. K. Kashihara, The diophantine equation $$x^2-1=(y^2-1)(z^2-1)$$,
Res. Rep. Anan College Tech. No. 26 (1990), 119-130.
2. Shin-Ichi Katayama and Kenji Kashihara, On the structure of the Integer Solutions of $$z^2=(x^2-1)(y^2-1)+a$$
J. Math. Tokushima Univ. Vol 24 (1990), 1-11.
The diagram on the last page of [1] is produced according to the 2-branched formulae: $\begin{array}{ccc} (x,y,z) & \rightarrow & (xy+z(y^2-1), y, x+zy)\\ & \searrow & \\ & & (xz+y(z^2-1), x+zy,z), \end{array}$ where $$x^2-1=(y^2-1)(z^2-1)$$.