Which numbers N can be written as pq(p+q) with p<q, in at least two different ways?

Still thinking about this. In the meantime here is a truly remarkable number:

903210 = [69,85] = [51,110] = [46,119] = [23,187] = [15,238] = [6,385].

where [p,q] = pq(p+q) and p,q coprime.

BTW, is this related to my recent tangent circles questions?

(note: answer edited to add one pair).

Edit: re tangent circles -- thanks gianlino -- I thought those numbers looked familiar.

There are embarassingly many solutions. Probably that is why no pattern is obvious. However, it is very easy to find Pell-like subsequences, as well as elliptic curves over Q that give rise to infinitely many rational solutions, which by clearing of fractions give integer solutions.

Here are some examples. As above [p,q] denotes pq(p+q).

From the solution you gave, namely [2, 3] = [1, 5] = 30, we wonder what other solutions might have a pair of the form [x, x+1]. If we try [x, x+1] = [a, 2x+1] which is your solution when x=2 and a=1 the algebra leads to:

x = (-1 + 2a + sqrt(1+8a^2)) / 2, and so we need to make 1+8a^2 a perfect square. This is Pell like, leading to a-values of 1, 6, 35, 204, 1189, 6930 ... defined recursively by a_0 = 1, a_1 = 1 and a_n = 6a_(n-1) - a_(n-2).

This gives infinitely many solutions to your problem, which start like:

[x, x+1] = [a, 2x+1] = x(x+1)(2x+1)

[2, 3] = [1, 5] = 30

[14, 15] = [6, 29] = 6090

[84, 85] = [35, 169] = 1206660

[492, 493] = [204, 985] = 238917660

[2870, 2871] = [1189, 5741] = 47304519570

[16730, 16731] = [6930, 33461] = 9366056129430

etc.

We can play this game in endlessly many ways. For example, a computer search for solutions with one pair of the form [x, x+1] finds:

[2, 3] = 30 = [1, 5]

[10, 11] = 2310 = [7, 15]

[14, 15] = 6090 = [6, 29]

[19, 20] = 14820 = [5, 52]

[22, 23] = 22770 = [9, 46]

[42, 43] = 153510 = [35, 51]

[55, 56] = 341880 = [5, 259]

[57, 58] = 380190 = [2, 435]

[66, 67] = 588126 = [57, 77]

[77, 78] = 930930 = [22, 195]

No need to have the form [x, x+1]. It just guarantees one pair is relatively prime.

Noting that [2, 3] = [1, 5] is of the form [2x, a] = [x, x+4]. Do the algebra and get x = -3 + a + sqrt(1 - 6a + 2a^2), so we need 1 - 6a + 2a^2 = perfect square. Again Pell like. We find x-values:

1, 4, 14, 31, 89, 188, 526, 1103, 3073, ...

and solutions:

[x,x+4] = [a, 2x]

[1, 5] = [3, 2] = 30

[31, 35] = [15, 62] = 71610

[89, 93] = [39, 178] =1506414

[1103, 1107] = [459, 2206] = 2698456410

[3073, 3077] = [1275, 6146] = 58152069150

etc. (where I have weeded out some non-relatively prime solutions).

Finally, an elliptic curve family of solutions.

Looking at the solution [57, 58] = 380190 = [2, 435] above suggests considering [x,x+1] = [2,a]

Solving for a gives: a = -1 + sqrt(4x^3 + 6x^2 + 2x + 4) / 2. Thus x needs to be a positive x-coordinate of a rational point on the elliptic curve y^2 = 4x^3 + 6x^2 + 2x + 4. This turns out to be an elliptic curve of rank 3 and so has infinitely many rational points which can be generated at will. Some that are integral give solutions:

[x, x+1] = [2, a]

[10, 11] = [2, 33] = 2310

[15, 16] = [2, 60] = 7440

[16, 17] = [2, 66] = 8976

[57, 58] = [2, 435] = 380190

[207, 208] = [2, 2988] = 17868240

A sample non-integer rational point has x-coordinate 656/529, giving a rational solution:

[656/529, 1185/529] = [2, 17220/12167] = 1431119760 / 148035889, which can be multiplied by 23^3 to give the integer solution: [15088, 27255] = [24334, 17220] = 17412434119920. Note that the first pair has gcd=23 and the second pair has gcd=2. So, although these are not relatively prime the equality cannot be reduced to one with smaller numbers.

I'm sure this is just the tip of the iceberg, but it is enough for one problem.

Edit2: I have corrected some typos above which would have made it confusing.

Do you mean p and q are both primes? Else N = 6 should be the smallest solution, 6 = 1*2 (1+2)

EDIT: Oh! sorry

EDIT2:

240 = 2*10(2+10) = 4*6(4+6)

390 = 2*13(2+13) = 3*10(3+10) << p, q coprime

EDIT3::

810 = 3*15(3+15) = 6*9(6+9)

880 = 2*20(2+20) = 5*11(5+11) << p, q coprime

1008 = 4*14(4+14) = 7*9(7+9)

1020 = 3*17(3+17) = 5*12(5+12) << p, q coprime

1680 = 5*16(5+16) = 6*14(6+14)

1920 = 2*30(2+30) = 4*20(4+20) = 8*12(8+12)

2100 = 3*25(3+25) = 4*21(4+21) << p, q coprime

2310 = 2*33(2+33) = 7*15(7+15) = 10*11(10+11) << p, q coprime

2484 = 8*14(8+14) = 4*23(4+23)

2970 = 3*30(3+30) = 5*22(5+22)

3120 = 4*26(4+26) = 6*20(6+10)

3360 = 2*40(2+40) = 3*32(3+32) = 10*14(10+14)

4320 = 5*27(5+27) = 6*24(6+24)

4914 = 3*39(3+39) = 13*14(13+14)

5460 = 4*35(4+35) = 13*15(13+15) << p, q coprime

5670 = 3*42(3+42) = 9*21(9+21)

6090 = 6*29(6+29) = 14*15(14+15) << p, q coprime

6270 = 2*55(2+55) = 5*33(5+33) = 11*19(11+19) << p, q coprime

6480 = 3*45(3+45) = 6*30(6+30) = 12*18(12+18)

6630 = 5*34(5+34) = 13*17(13+17) << p, q coprime

Still searching for a pattern.

EDIT4: for N < 10000

N = 30, 240, 390, 810, 880, 1008, 1020, 1680, 1920, 2100, 2310, 2484, 2970, 3120, 3360, 4320, 4914, 5460, 5670, 6090, 6270, 6480, 7040, 7380, 7440, 7770, 8064, 8160, 8190, 8190, 8448, 8580, 8976, 9240, 9520

but for N < 10000, with p, q coprime

N = 30, 390, 880, 1020, 2100, 2310, 5460, 6090, 6270, 6630, 7380, 7770, 8190, 8580, 9240

EDIT5: for 10000 <= N < 20000, with p, q coprime

N = 12210, 12432, 13110, 14280, 14820, 16770, 17670, 18480, 19740

Still not seeing a pattern here.