Problems & Puzzles: Puzzles

 Puzzle 378. Sequences embedded in decimal periods of fractions Luis Rodríguez sends the following nice puzzle. I don't remember where I saw that 1000/998999 contains a part of the Fibonacci sequence 0.001001002003005008013021034055089144233... I decided to search for fractions that could represent other familiar sequences. By example the squares are represented by 10100/970299 = 0.0104091625364964.. The triangular numbers: 10000/970299 = 0.010306101521283645556678.. It produces 16 triangular numbers. The powers of 3 (3^n): 10/9997 = 0.0100300902708124372921876561.. Now I propose to search for the fraction that reproduce the primes, in such a manner that the the number of digits of decimal with primes, be greater than the number of digits of denominator +  numerator. I have found that: 19893/979706 = 0.020305071113172.. 5 + 6 = 11 < 14. Q. Find a decimal with primes beyond 17 from a fraction whose number of digits: numerator + denominator be less than the number of decimals?

Contributions came from Jaroslaw Wroblewski, Jacques Tramu, J. K. Andersen & Nick McGrath. Before here you all have a useful link for this puzzle:

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Jarek found too many small & large solutions, I will just copy one small and his largest:

16740062 / 824427647 =
0.0203050711131719239
8 + 9 = 17 < 18

...

p/q =

0.00020003000500070011...1699170917211723173317411
539 + 543 = 1082 < 1084

> a) can you explain your method?

I am taking the number I want to approximate, e.g.
0.020305...838997

Then I expand it as continued fraction, which gives me good rational
approximations of the given number....how continued fractions give rational approximations is a standard stuff...

After I get a good approximation, I have to check whether the fraction
has less digits than used in primes generated by decimal extension, as
required by the puzzle formulation.

> b) can you control the production of the integers p/q in the fraction in
> order to get prime numbers at least for one of them?

I have no influence on the p/q I produce. Perhaps I could produce a bit
more of the p/q's, but not by much. From the number of p/q's that are
being produced, I feel that there should be examples with one of them
prime, but very very rare. It is uncertain whether any such example can
be found. It is highly unlikely both p and q can be prime at the same time. I will let you know if I can find an example with prime p or q.

...

Searching the files I had mailed you I have found that the fraction:

199155200426057675549 / 9808150846458521632654 =
0.020305071113171923293137414347535961677173794
21 + 22 = 43 < 44

has prime numerator p and the fraction

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Jacques wrote:

16740062/824427647 =  0.020305071113171923....
8 + 9 = 17 < 18

***

JKA wrote:

The below decimals end with the first incorrect digit.

16740062 / 824427647 = 0.0203050711131719239... 8 + 9 = 17 < 18

80249675696040019 / 3952198701928306021 =
0.020305071113171923293137414347535961674... 17 + 19 = 36 < 38

5682681272389079184061 / 279865125352982251832018 =
0.02030507111317192329313741434753596167717379838996... 22 + 24 = 46 < 49

Computed trivially with the PARI/GP function bestappr. It uses continued fractions.
I stopped before 3-digit primes because it was unclear which decimals to use.
If all primes from 2 are concatenated with no 0's between,
then here is perhaps the first case with 2 extra digits:

6118475199813191932767364007023662474929750854787196686115120528196527776 /
25957514841094595750843280466625657978907866654579733000433842604026681753 =
0.23571113171923293137414347535961677173798389971011031071091131271311371391\
4915115716316717317918119119319719921122322722923323924125125726326927127720...
73 + 74 = 147 < 149

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Nick wrote:

I wrote a little program using continued fractions to get the best rational approximation. There seem to be many solutions for which the number of digits in the fraction is equal to the number of decimals but very few for which the number of digits is less.
The next example higher than the one given is:

133787802897360/6588886202450761= 0.02030507111317192329313741434753..

where #digits is 31 and #decimals is 32.

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