Wikipedia:Reference desk/Archives/Mathematics/2018 January 16

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January 16

Three integers optic equation producing an integer

In what condition does the sum of multiplicative inverses of three integers gives also the inverse of an integer as an extension of optic equation with 2 integer variables? (Semi-protected edit request by 5.2.200.163 (talk) 16:12, 16 January 2018 (UTC); fulfilled by ToE 16:38, 16 January 2018 (UTC))

See Egyptian fraction. Count Iblis (talk) 17:13, 16 January 2018 (UTC)

Working purely from analogy with Optic equation#Solution, I find that ${\frac {1}{a}}+{\frac {1}{b}}+{\frac {1}{c}}={\frac {1}{d}}$ is satisfied by $a=km(mn+mp+np),\quad b=kn(mn+mp+np),\quad c=kp(mn+mp+np),\quad d=kmnp$.

Hopefully someone who is better with Diophantine equations can tell you if this yields all solutions. -- ToE 17:22, 16 January 2018 (UTC)

I'm pretty sure this does not include the solution 1/4+1/4+1/2=1/1. It is fairly easy to show that all rational solutions to 1/x+1/y+1/z=1 are given by x=(p+q+r)/p, y=(p+q+r)/q, z=(p+q+r)/r where p, q, r are integers with no common factors. So you can get all solutions of the original equation by taking a=qr(p+q+r), b=pr(p+q+r), c=pq(p+q+r), d=pqr, reducing to lowest terms, and multiplying by a constant. For example, with p=1, q=1, r=2, you get a=8, b=8, c=4, d=2, reducing to lowest terms gives a=4, b=4, c=2, d=1, and multiplying by a constant produces a=4k, b=4k, c=2k, d=k. This is more or less what happens in the two integer case except that there the reduction to lowest terms step is automatic. The two variable case reduces to finding rational points on a rational quadratic curve, but for the three variable case you get a rational cubic surface and the analysis is trickier. --RDBury (talk) 20:30, 16 January 2018 (UTC)

RDBury, what about e.g. 1/18+1/9+1/3=1/2? I'm not sure your method can produce that one. 93.136.7.174 (talk) 03:40, 17 January 2018 (UTC)

With p=1, q=2, r=6, you get a=108, b=54, c=18, d=12, reducing to lowest terms gives a=18, b=9, c=3, d=2. -- ToE 07:21, 17 January 2018 (UTC)

Actually ToE's equations above do give all solutions provided you allow k to be a non-integer. For example, if you take m=n=2, and p=1 you get mn+mp+np=8, a=16k, b=16k, c=8k, d=4k, and then the solution 1/4+1/4+1/2=1/1 pops out when you take k=1/4. In both approaches there is the awkward step of finding common factors, and this makes it more difficult to do something practical like list all solutions within a given bound. As a side note, the case d=1 is well-known because the solutions correspond to certain uniform tilings of the plane; specifically a=b=c=3 to the Hexagonal tiling, a=b=4, c=2 to the Truncated square tiling, and a=6, b=3, c=2 to the Truncated trihexagonal tiling. Allowing for permutations, there are 10 solutions for d=1, and a natural question is whether there are always a finite number of solutions for a given d. --RDBury (talk) 01:03, 18 January 2018 (UTC)

There is always a finite number of solutions for given d: the largest of a, b, c is at most 3d, while the middle value is at most 2d(d + 1) or something like that (because the difference 1/d - 1/a is at least 1/d - 1/(d + 1)). --JBL (talk) 03:05, 18 January 2018 (UTC)

The number of solutions for a given d is given by OEIS: A004194. According one of the links given there, a complete solution will be difficult because it would solve the Erdős–Straus conjecture. --RDBury (talk) 09:47, 18 January 2018 (UTC)

How is the situation in the next case with for four integers a, b, c, d reciprocals to give a fifth reciprocal of an integer e comparatively to the case with 3 integers? Is there some recurrence relation between cases?--5.2.200.163 (talk) 17:16, 23 January 2018 (UTC)

The pattern from the optic equation and the above n=3 case leads me to conjecture this:

Let $C=\sum _{j=1}^{n}{\frac {\prod _{p=1}^{n}b_{p}}{b_{j}}}$ for any natural numbers $b_{p}$ and $n.$ Then if

$a_{i}=b_{i}C$

for i = 1,...,n and if

$a_{n+1}=\prod _{p=1}^{n}b_{p}$

we have $1/a_{1}+\dots +1/a_{n}=1/a_{n+1}.$

Moreover, I conjecture that this gives all solutions if these are multiplied by k as k runs over the positive rationals that give integer solutions. (But I haven’t had time to check this.) Loraof (talk) 22:45, 23 January 2018 (UTC)