20221031, 07:23  #1 
Jul 2015
26_{16} Posts 
November 2022

20221031, 08:25  #2 
Aug 2022
China
2^{5} Posts 
.
Last fiddled with by SuikaPredator on 20221031 at 08:26 Reason: deleted 
20221031, 08:50  #3 
Romulan Interpreter
"name field"
Jun 2011
Thailand
10100000101001_{2} Posts 
Not really. While, with the current computing power of a good CPU today, a single line in Pari incrementing the enumerator and denominator alternatively (the sillystupid way) will give you the EP100 solution in about half hour or so (didn't have the patience and the time, stopped it after few seconds), here (by replacing 2 with the constant they give) it will not work, due to the size of the numbers, you will get quite old going one by one for 100 digits , and will have to do binary search, or come with a more clever algorithm.
Code:
gp > m=1; n=1; while(1, if((zn=(2*n*(n1)))<(zm=(m*(m1))),n++,if(zn>zm,m++,print(n", "m);n++))) 1, 1 3, 4 15, 21 85, 120 493, 697 2871, 4060 16731, 23661 97513, 137904 568345, 803761 3312555, 4684660 19306983, 27304197 112529341, 159140520 Last fiddled with by LaurV on 20221031 at 15:02 
20221101, 03:15  #4 
Sep 2022
3×23 Posts 
a must be one of the following forms mod 487085 (i think), but considering the fact we are dealing with 100 digit numbers it hardly helps:
0,1,79850,114985,194835,292251,372101,407236. Last fiddled with by Rubiksmath on 20221101 at 03:22 
20221101, 05:58  #5 
Oct 2017
213_{8} Posts 

20221101, 06:06  #6 
Sep 2022
3×23 Posts 

20221101, 08:21  #7 
Aug 2022
China
2^{5} Posts 
I'm sorry, but I discovered my mistake immediately after posting that one. Actually I misread the fraction 1/974170 in the problem statement into 1/2. Maybe the actual problem could also be solved by continued fractions but with some more complex deductions than the 1/2 version.

20221102, 06:52  #8 
Jul 2015
2×19 Posts 
Please explain
"Your goal: find a,b such that the probability for two comfortable socks is exactly \frac{1}{974170} , and such that this is the minimal solution with b having at least 100 digits (minimal with respect to the size of b )"
My understanding is b should be bigger than 10^99 and smaller than any other potential b? 
20221102, 13:28  #9  
Feb 2017
Nowhere
3^{4}·7·11 Posts 
Quote:
If I did my sums correctly, the smallest possible b is 114985, with a = 117. (I don't think this is giving away anything useful for answering the stated question.) 

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