Interesting Math Problem

Interesting Math Problem

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Arithmetic Progressions of Rational Squares

There exist infinitely many 3-element arithmetic progressions of square integers, such as these:
(1 + 24 = 25 + 24 = 49)
(49 + 120 = 169 + 120 = 289)
(49 + 140 = 289 + 140 = 529)
(289 + 336 = 625 + 336 = 961)
(961 + 720 = 1681 + 720 = 2401)
(2209 + 2016 = 4225 + 2016 = 6241)
(2209 + 122400 = 124609 + 122400 = 247009)
(82369 + 15600 = 97969 + 15600 = 113569)
(12769 + 100800 = 113569 + 100800 = 214369)

When the common difference is divisible by smaller square integers,
there exist 3-element arithmetic progressions where the integers above
are the numerators, and the divisor, a square, is the denominator.
In other words, arithmetic progressions of rational numbers, with common
difference an integer.

Very few differences have more than one progression,
at least in the range of numbers I searched.
6, 138, 154, and 210 are among them.

In other cases, the sequences are related, with just a change of denominator.
If (p/q)^2 is a middle number, then you also can have
(p/(any divisor of q))^2 as well. (Such as 65/12, 65/6, 65/4, 65/3.)
(337/120) is the basis of many progressions, since 120 has many divisors.

It's interesting to note that (65/4)^2 shows up twice
as a middle number: for difference 126 and for difference 231.

This table show some of these progressions, up to common difference 250.

The first column shows the common difference.
The second column shows the common difference as a fraction.
The third column shows the square roots of the numbers (squared).
The fourth column shows the three numbers in p/q form.

ą 5	(31/12)˛, (41/12)˛, (49/12)˛	720/144	(961/144, 1681/144, 2401/144)
ą 6	(1/2)˛, (5/2)˛, (7/2)˛	24/4	(1/4, 25/4, 49/4)
ą 6	(1151/140)˛, (1201/140)˛, (1249/140)˛	117600/19600	(1324801/19600, 1442401/19600, 1560001/19600)
ą 7	(113/120)˛, (337/120)˛, (463/120)˛	100800/14400	(12769/14400, 113569/14400, 214369/14400)
ą 14	(47/12)˛, (65/12)˛, (79/12)˛	2016/144	(2209/144, 4225/144, 6241/144)
ą 15	(7/4)˛, (17/4)˛, (23/4)˛	240/16	(49/16, 289/16, 529/16)
ą 20	(31/6)˛, (41/6)˛, (49/6)˛	720/36	(961/36, 1681/36, 2401/36)
ą 21	(17/4)˛, (25/4)˛, (31/4)˛	336/16	(289/16, 625/16, 961/16)
ą 24	(1151/70)˛, (1201/70)˛, (1249/70)˛	117600/4900	(1324801/4900, 1442401/4900, 1560001/4900)
ą 24	(1/1)˛, (5/1)˛, (7/1)˛	24/1	(1/1, 25/1, 49/1)
ą 28	(113/60)˛, (337/60)˛, (463/60)˛	100800/3600	(12769/3600, 113569/3600, 214369/3600)
ą 30	(7/2)˛, (13/2)˛, (17/2)˛	120/4	(49/4, 169/4, 289/4)
ą 34	(47/60)˛, (353/60)˛, (497/60)˛	122400/3600	(2209/3600, 124609/3600, 247009/3600)
ą 39	(287/20)˛, (313/20)˛, (337/20)˛	15600/400	(82369/400, 97969/400, 113569/400)
ą 41	(431/120)˛, (881/120)˛, (1169/120)˛	590400/14400	(185761/14400, 776161/14400, 1366561/14400)
ą 45	(31/4)˛, (41/4)˛, (49/4)˛	720/16	(961/16, 1681/16, 2401/16)
ą 54	(3/2)˛, (15/2)˛, (21/2)˛	216/4	(9/4, 225/4, 441/4)
ą 56	(47/6)˛, (65/6)˛, (79/6)˛	2016/36	(2209/36, 4225/36, 6241/36)
ą 60	(7/2)˛, (17/2)˛, (23/2)˛	240/4	(49/4, 289/4, 529/4)
ą 63	(113/40)˛, (337/40)˛, (463/40)˛	100800/1600	(12769/1600, 113569/1600, 214369/1600)
ą 65	(7/12)˛, (97/12)˛, (137/12)˛	9360/144	(49/144, 9409/144, 18769/144)
ą 70	(17/6)˛, (53/6)˛, (73/6)˛	2520/36	(289/36, 2809/36, 5329/36)
ą 78	(623/30)˛, (677/30)˛, (727/30)˛	70200/900	(388129/900, 458329/900, 528529/900)
ą 80	(31/3)˛, (41/3)˛, (49/3)˛	720/9	(961/9, 1681/9, 2401/9)
ą 84	(17/2)˛, (25/2)˛, (31/2)˛	336/4	(289/4, 625/4, 961/4)
ą 96	(1151/35)˛, (1201/35)˛, (1249/35)˛	117600/1225	(1324801/1225, 1442401/1225, 1560001/1225)
ą 96	(2/1)˛, (10/1)˛, (14/1)˛	96/1	(4/1, 100/1, 196/1)
ą 111	(337/140)˛, (1513/140)˛, (2113/140)˛	2175600/19600	(113569/19600, 2289169/19600, 4464769/19600)
ą 112	(113/30)˛, (337/30)˛, (463/30)˛	100800/900	(12769/900, 113569/900, 214369/900)
ą 120	(7/1)˛, (13/1)˛, (17/1)˛	120/1	(49/1, 169/1, 289/1)
ą 125	(155/12)˛, (205/12)˛, (245/12)˛	18000/144	(24025/144, 42025/144, 60025/144)
ą 126	(47/4)˛, (65/4)˛, (79/4)˛	2016/16	(2209/16, 4225/16, 6241/16)
ą 135	(21/4)˛, (51/4)˛, (69/4)˛	2160/16	(441/16, 2601/16, 4761/16)
ą 136	(47/30)˛, (353/30)˛, (497/30)˛	122400/900	(2209/900, 124609/900, 247009/900)
ą 138	(521/30)˛, (629/30)˛, (721/30)˛	124200/900	(271441/900, 395641/900, 519841/900)
ą 138	(527/20)˛, (577/20)˛, (623/20)˛	55200/400	(277729/400, 332929/400, 388129/400)
ą 141	(2207/56)˛, (2305/56)˛, (2399/56)˛	442176/3136	(4870849/3136, 5313025/3136, 5755201/3136)
ą 145	(719/84)˛, (1241/84)˛, (1601/84)˛	1023120/7056	(516961/7056, 1540081/7056, 2563201/7056)
ą 150	(5/2)˛, (25/2)˛, (35/2)˛	600/4	(25/4, 625/4, 1225/4)
ą 150	(1151/28)˛, (1201/28)˛, (1249/28)˛	117600/784	(1324801/784, 1442401/784, 1560001/784)
ą 154	(23/30)˛, (373/30)˛, (527/30)˛	138600/900	(529/900, 139129/900, 277729/900)
ą 154	(41/6)˛, (85/6)˛, (113/6)˛	5544/36	(1681/36, 7225/36, 12769/36)
ą 156	(287/10)˛, (313/10)˛, (337/10)˛	15600/100	(82369/100, 97969/100, 113569/100)
ą 161	(17/24)˛, (305/24)˛, (431/24)˛	92736/576	(289/576, 93025/576, 185761/576)
ą 164	(431/60)˛, (881/60)˛, (1169/60)˛	590400/3600	(185761/3600, 776161/3600, 1366561/3600)
ą 165	(217/24)˛, (377/24)˛, (487/24)˛	95040/576	(47089/576, 142129/576, 237169/576)
ą 174	(617/30)˛, (733/30)˛, (833/30)˛	156600/900	(380689/900, 537289/900, 693889/900)
ą 175	(113/24)˛, (337/24)˛, (463/24)˛	100800/576	(12769/576, 113569/576, 214369/576)
ą 180	(31/2)˛, (41/2)˛, (49/2)˛	720/4	(961/4, 1681/4, 2401/4)
ą 189	(51/4)˛, (75/4)˛, (93/4)˛	3024/16	(2601/16, 5625/16, 8649/16)
ą 190	(161/6)˛, (181/6)˛, (199/6)˛	6840/36	(25921/36, 32761/36, 39601/36)
ą 210	(1/2)˛, (29/2)˛, (41/2)˛	840/4	(1/4, 841/4, 1681/4)
ą 210	(23/2)˛, (37/2)˛, (47/2)˛	840/4	(529/4, 1369/4, 2209/4)
ą 210	(97/4)˛, (113/4)˛, (127/4)˛	3360/16	(9409/16, 12769/16, 16129/16)
ą 216	(3/1)˛, (15/1)˛, (21/1)˛	216/1	(9/1, 225/1, 441/1)
ą 221	(49/12)˛, (185/12)˛, (257/12)˛	31824/144	(2401/144, 34225/144, 66049/144)
ą 224	(47/3)˛, (65/3)˛, (79/3)˛	2016/9	(2209/9, 4225/9, 6241/9)
ą 231	(23/4)˛, (65/4)˛, (89/4)˛	3696/16	(529/16, 4225/16, 7921/16)
ą 240	(7/1)˛, (17/1)˛, (23/1)˛	240/1	(49/1, 289/1, 529/1)
ą 245	(217/12)˛, (287/12)˛, (343/12)˛	35280/144	(47089/144, 82369/144, 117649/144)