5
$\begingroup$

Find all positive integers $k,m,n$ satisfying: $\frac1k+\frac1m+\frac1n=\frac{1}{1996}$

The trivial answer is: $k=m=n=3*1996$
$kmn=1996(km+mn+nk)=499\times4\times(km+mn+nk)$ , now $kmn$ must be divisible by all factors in RHS , but how can we get more info about $k,m,n$ ?

$\endgroup$
  • 2
    $\begingroup$ $\frac{1}{1996}=\frac{1}{1997}+\frac{1}{3986012}=\frac{1}{1998}+\frac{1}{1994004}=\frac{1}{1999}+\frac{1}{1330002}+\frac{1}{2653356650004}$...There are many possible solutions. $\endgroup$ – Mc Cheng Jul 11 '16 at 15:38
  • 6
    $\begingroup$ For example, $\frac{1}{x}=\frac{1}{x+1}+\frac{1}{x(x+1)}$ $\endgroup$ – Mc Cheng Jul 11 '16 at 15:42
  • 3
    $\begingroup$ math.stackexchange.com/questions/450280/… $\endgroup$ – individ Jul 11 '16 at 15:43
  • 2
    $\begingroup$ It is easy to write $\frac{1}{x}=\frac{1}{x+503}+\frac{503}{x(x+503)}$. In this case $x=1996$, so $x(x+503)=4988004$. After trial and errors, we may see that $4988004=499*9996$, arriving to your answer. $\endgroup$ – Mc Cheng Jul 11 '16 at 16:04
  • 3
    $\begingroup$ You can also use $\frac{1}{2}+\frac{1}{3}+\frac{1}{6}=1$ and $\frac{1}{2}+\frac{1}{4}+\frac{4}=1$ to get other answers. $\endgroup$ – Thomas Andrews Jul 11 '16 at 16:18
4
$\begingroup$

Solutions only for $k=m$.

Assume $k=m$, then $\frac{1}{1996}=\frac{2n+m}{mn}$ or $mn=1996\cdot 2n + 1996m$ or $$mn - 1996\cdot 2n - 1996\cdot m =0$$

or $$(n-1996)(m-1996\cdot 2)=2\cdot 1996^2$$

or:

$$(n-1996)(m-1996\cdot 2)=2\cdot 1996^2.$$

If $d\mid 2\cdot 1996^2$ then $n=1996+d$ and $m-1996\cdot 2 = \frac{2\cdot 1996^2}{d}$ or $m=1996\cdot 2 + \frac{2\cdot 1996^2}{d}$.

Since $1996=2^2\cdot 499$, then $d\mid 2^5\cdot 499^2$, which yields already 18 possible solutions.

The extreme case, when $d=1$, gives $n=1997$ and $m=k=2\cdot 1996 \cdot 1997$. When $d=2\cdot 1996^2$, $n=1996(1+2\cdot 1996)$ and $m=k=1996\cdot 2 + 1$.

Algorithm for finding all soluions

If you assume $k<m,n$, then $1996<k<3\cdot 1996$, and you want:

$$\frac{k-1996}{1996k}=\frac{1}m+\frac1n$$

Or $$(k-1996)^2mn - 1996k(k-1996)(m+n)$$

or $$\left((k-1996)m-1996k\right)\left((k-1996)n-1996k\right)=1996^2k^2$$

So you need to find a divisor $d\mid 1996^2k^2$ where both $k-1996\mid d+1996k$ and $k-1996\mid \frac{1996^2k^2}{d}+1996k$, or if:

$$k-1996\mid d+1996^2\\k-1996\mid \frac{1996^2k^2}{d}+1996^2$$

We could write a computer program to enumerate such pairs $(k,d)$ pretty easily, and then solve for $m,n$.

My ruby script turned up 1517 answers, assuming $k\leq m\leq n$, which turns out to be too large for Stack Exchange to handle as raw data, but here is the complete list of $(k,m)$ pairs.

1997 3986013
1997 3986014
1997 3986016
1997 3986020
1997 3986028
1997 3986511
1997 3987010
1997 3988008
1997 3990004
1997 3993996
1997 4235013
1997 4484014
1997 4982016
1997 5978020
1997 7970028
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1997 3990006
1997 3994000
1997 4001988
1997 4017964
1997 4982515
1997 5979018
1997 7972024
1998 1994005
1998 1994006
1998 1994008
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1998 1994503
1998 1995002
1998 1996000
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1998 2001988
1998 2243005
1998 2492006
1998 2990008
1998 3986012
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1998 1994028
1998 1994052
1998 1995501
1998 1996998
1998 1999992
1998 2005980
1998 2017956
1998 2741007
1998 3488010
1998 1994013
1998 1994022
1998 1994040
1998 1994076
1998 1994148
1998 1998495
1998 2002986
1998 2011968
1998 2029932
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1998 1994436
1998 2007477
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1998 2964060
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1998 2880228
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1998 3489507
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1998 1999480
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1998 2015908
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1998 3360266
1998 1998111
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1998 2010432
1998 2026860
1998 2059716
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1998 2881116
1998 3768228
1998 2326671
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1998 3324672
1998 2992005
1999 1330002
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1999 1330334
1999 1331332
1999 1496002
1999 1994004
1999 1331334
1999 1335332
1999 1995002
2000 998001
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2000 998016
2000 998032
2000 998064
2000 998128
2000 998256
2000 998499
2000 998998
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2000 1001992
2000 1005984
2000 1013968
2000 1029936
2000 1061872
2000 1125744
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2000 998005
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2363 12852
2380 12475
2380 17465
2380 12376
2380 12495
2385 12240
2394 13972
2394 12012
2396 11976
2396 11980
2397 11976
2400 11976
2400 12000
2400 12475
2408 11976
2412 11573
2412 11574
2413 11550
2418 11440
2424 11976
2424 11312
2430 11178
2432 11136
2432 11134
2434 11092
2440 10978
2440 10980
2442 10978
2442 10989
2448 10812
2450 10780
2453 10978
2460 10590
2465 10540
2466 10479
2478 10262
2478 10266
2480 10230
2484 10160
2484 10212
2495 9981
2495 9982
2495 9984
2495 9988
2495 9996
2495 10479
2495 10978
2495 11976
2495 13972
2495 17964
2495 9985
2495 9990
2495 10000
2495 10020
2495 10060
2495 12475
2495 14970
2495 19960
2495 10005
2495 10030
2495 10080
2495 10180
2495 10380
2496 9980
2496 9984
2497 9988
2499 9996
2500 9980
2500 10000
2505 10020
2511 9732
2515 10060
2520 9600
2520 9980
2520 9630
2530 9460
2532 9429
2534 9412
2541 9317
2548 9282
2558 9085
2560 9060
2568 8982
2568 8988
2574 8982
2574 8892
2576 8865
2580 8818
2580 8820
2580 14970
2584 8772
2595 8650
2604 10479
2604 8556
2620 8384
2624 8340
2625 8330
2625 8400
2626 8320
2628 8982
2630 8280
2640 8184
2640 8250
2646 8127
2662 7984
2662 7986
2664 7984
2664 7992
2672 7984
2672 8016
2695 7700
2703 7632
2704 7984
2710 7588
2717 7524
2722 7485
2724 7485
2724 7491
2727 7474
2728 7440
2730 7485
2736 7380
2737 7378
2740 7485
2745 7320
2754 7252
2769 7150
2788 7038
2788 8483
2789 7020
2796 6986
2796 6990
2800 6986
2800 7000
2808 6912
2808 6903
2814 6986
2816 6864
2820 6831
2830 6773
2839 6722
2852 6670
2856 6630
2856 6664
2872 6544
2884 6487
2884 6489
2886 6487
2890 6460
2898 6417
2904 6384
2924 6321
2937 6230
2964 6162
2968 6095
2970 6105
2972 6078
2980 6045
2994 5989
2994 5990
2994 5992
2994 5996
2994 6004
2994 6487
2994 6986
2994 7984
2994 9980
2994 5991
2994 5994
2994 6000
2994 6012
2994 6036
2994 7485
2994 8982
2994 11976
2994 5997
2994 6006
2994 6024
2994 6060
2994 6132
2994 10479
2995 5988
2995 5990
2996 5988
2996 5992
2997 5988
2997 5994
2998 5988
2998 5996
3000 5988
3000 6000
3002 6004
3003 5988
3006 5988
3006 6012
3012 5988
3016 5902
3018 6036
3020 5889
3021 5883
3024 5872
3030 5988
3069 5709
3081 5668
3105 5589
3117 5550
3149 5452
3180 5361
3200 5305
3213 5292
3234 5236
3250 5200
3252 5168
3276 5109
3280 5125
3281 5100
3282 5094
3283 5092
3290 5076
3310 5028
3327 4990
3328 4990
3328 4992
3330 4990
3330 4995
3332 4990
3332 4998
3335 4990
3340 4990
3340 5010
3342 4956
3360 4990
3367 4914
3380 4875
3390 4854
3402 4830
3423 4788
3448 4741
3456 4736
3456 4725
3460 4990
3493 4658
3493 4660
3493 4990
3493 5988
3493 4662
3493 4676
3493 6986
3493 4690
3493 4788
3495 4660
3500 4645
3507 4676
3512 4624
3556 4550
3582 4508
3591 4494
3593 4491
3594 4491
3596 4491
3596 4495
3600 4491
3600 4480
3600 4500
3609 4491
3610 4465
3636 4491
3640 4420
3648 4408
3660 4392
3682 4359
3685 4355
3696 4340
3708 4326
3745 4280
3748 4270
3780 4230
3824 4176
3840 4160
3852 4491
3870 4122
3920 4080
3930 4056
3938 4048
3948 4037
3960 4026
3982 4004
3992 3993
3992 3994
3992 3996
3992 4000
3992 4008
3992 4024
3992 4056
3992 4491
3992 4990
3992 5988
3992 7984
3993 3993
3994 3994
3996 3996
4000 4000
4008 4008
4024 4024
4491 5988
4491 4491
4990 5988
4990 4990
5988 5988
$\endgroup$
  • $\begingroup$ Good practice(+1),but how to extend it to general case? $\endgroup$ – Hamid Reza Ebrahimi Jul 11 '16 at 17:04
  • 2
    $\begingroup$ My script takes less than 2 seconds to run. $\endgroup$ – Thomas Andrews Jul 11 '16 at 19:07

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