Re: Three rational triples

"Keith F. Lynch" <kfl@KeithLynch.net> writes:

HenHanna <HenHanna@dev.null> wrote:

Keith F. Lynch wrote:

Since it's been more than a week, and nobody has figured it out:
Each of them has a sum that's equal to its product and is an integer.

>

i think one person said exactly that.

>
Who and when?  I didn't see any such post.
>

Is it easy to find them?

>
No, even though there are infinitely many.  Try and find one I
didn't list.
>
Constraints:  All three numbers must be positive, real, and rational,
but not integers.

They're literally everywhere. Given 2 rational numbers, there's a solution

(Here I've capped numerators and denominators to 50.)

$ echo 'for(n=1,10000,a=(1+random(49))/(2+random(48));if(denominator(a)==1,next);b=1/a+(1+random(50))/(2+random(50));if(denominator(b)==1||numerator(b)>=50,next);x=(a+b)/(a*b-1);if(denominator(x)>1&&numerator(x)<50&&denominator(x)<50,print(a" "b" "x" "a+b+x"="a*b*x)))' | gp -q > crap
$ sort -n crap | uniq
1/2 11/4 26/3 143/12=143/12
1/2 16/5 37/6 148/15=148/15
1/2 19/8 46/3 437/24=437/24
1/2 26/3 11/4 143/12=143/12
1/2 29/7 13/3 377/42=377/42
1/2 8/3 19/2 38/3=38/3
1/3 21/5 34/3 238/15=238/15
1/3 25/7 41/2 1025/42=1025/42
1/3 31/5 49/8 1519/120=1519/120
1/3 36/7 23/3 92/7=92/7
1/4 11/2 46/3 253/12=253/12
2/3 11/2 37/16 407/48=407/48
2/3 11/6 45/4 55/4=55/4
2/3 7/2 25/8 175/24=175/24
2/3 9/4 35/6 35/4=35/4
2/5 7/2 39/4 273/20=273/20
3/2 13/7 47/25 1833/350=1833/350
3/2 17/3 43/45 731/90=731/90
3/2 29/24 10/3 145/24=145/24
3/2 3/2 12/5 27/5=27/5
3/2 4/3 17/6 17/3=17/3
3/2 5/3 19/9 95/18=95/18
3/2 8/3 25/18 50/9=50/9
3/4 19/3 17/9 323/36=323/36
3/4 8/3 41/12 41/6=41/6
3/5 11/3 32/9 352/45=352/45
4/3 12/11 16/3 256/33=256/33
4/3 13/2 47/46 611/69=611/69
4/3 13/4 11/8 143/24=143/24
4/3 17/6 3/2 17/3=17/3
4/3 49/12 39/32 637/96=637/96
4/3 7/4 37/16 259/48=259/48
4/3 7/6 9/2 7=7
4/3 9/4 43/24 43/8=43/8
4/5 7/3 47/13 1316/195=1316/195
5/2 7/5 39/25 273/50=273/50
5/3 8/5 49/25 392/75=392/75
6/5 3/2 27/8 243/40=243/40
6/7 19/12 41/6 779/84=779/84
6/7 9/2 15/8 405/56=405/56
7/2 5/3 31/29 1085/174=1085/174
7/3 9/7 38/21 38/7=38/7
7/9 29/7 31/14 899/126=899/126
8/5 3/2 31/14 186/35=186/35
9/2 5/3 37/39 185/26=185/26
15/7 3/5 48/5 432/35=432/35
20/21 5/2 5/2 125/21=125/21
22/7 10/11 24/11 480/77=480/77
42/11 20/21 38/21 1520/231=1520/231

I'm particularly enamoured with this find:
4/3 7/6 9/2 7=7

Widening the net, I was unable to find any other integer sum/products.
Can anyone else find another one - I suspect actually using some algebra
might make sense, rather than just probing randomly.

Other small denominators are everywhere:

1/12 63/4 152/3 133/2=133/2
1/28 147/4 824/7 309/2=309/2
1/3 9/2 29/3 29/2=29/2
1/33 729/22 24079/3 16119/2=16119/2
1/5 29/5 75/2 87/2=87/2
1/7 49/2 69/7 69/2=69/2

1/15 78/5 1175/3 1222/3=1222/3
1/18 172/9 621/2 989/3=989/3
1/2 16/3 7/2 28/3=28/3
1/2 19/2 8/3 38/3=38/3
1/2 7/2 16/3 28/3=28/3
1/4 16/3 67/4 67/3=67/3
1/4 19/4 80/3 95/3=95/3
1/5 25/3 64/5 64/3=64/3
1/5 64/5 25/3 64/3=64/3
1/6 15/2 92/3 115/3=115/3
1/6 20/3 123/2 205/3=205/3
1/7 55/7 196/3 220/3=220/3
1/78 169/2 13184/13 3296/3=3296/3
1/8 103/8 64/3 103/3=103/3
2/15 25/3 381/5 254/3=254/3
2/7 26/7 196/3 208/3=208/3
3/2 17/6 4/3 17/3=17/3
3/2 28/3 5/6 35/3=35/3
3/2 5/6 28/3 35/3=35/3
4/3 3/2 17/6 17/3=17/3
4/3 39/2 5/6 65/3=65/3
4/3 5/6 39/2 65/3=65/3
5/6 3/2 28/3 35/3=35/3
5/6 4/3 39/2 65/3=65/3




Phil
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