Problems & Puzzles: Puzzles

 

 

Problems & Puzzles: Puzzles

Puzzle 1254 Car 41, where are you?

 Here we go with a new puzzle, from another Curio from the pages of G.L. Honaker, Jr. 

 Consider a "prime race" based on the values of p mod 100 for each prime p. Over all primes there are 42 possible values of p mod 100 (1, 2, 3, 5, 7, 9, 11, 13, 17, 19, 21, 23, 27, 29, 31, 33, 37, 39, 41, 43, 47, 49, 51, 53, 57, 59, 61, 63, 67, 69, 71, 73, 77, 79, 81, 83, 87, 89, 91, 93, 97, or 99), but only 40 of these occur in the long term (since 2 and 5 are special cases). In this prime race we scan through the primes from the beginning, and keep track of how many primes there are having each of these 40 possible values for p mod 100, and we ask: when does each of the 40 racers "take the lead" (be the only one to have the largest count) for the first time? The first 39 first-time leaders that occur are, in order, 3, 7, 31, 57, 9, 83, 67, 19, 51, 23, 39, 37, 21, 87, 59, 73, 93, 97, 71, 33, 27, 53, 1, 91, 63, 17, 11, 69, 79, 49, 77, 47, 13, 81, 89, 61, 99, 29, 43 and they take the lead for the first time at primes 3, 307, 431, 857, 1109, 1583, 3967, 5519, 7451, 8423, 12739, 13337, 13921, 16087, 23059, 25873, 60793, 63997, 171671, 253433, 340127, 457553, 580201, 977791, 1267663, 1329217, 2059711, 2607469, 3032279, 6253549, 11761777, 34929547, 45740213, 51656281, 213350989, 227710961, 316867699, 1033090529, 11342219743, respectively. Every racer has been in the lead except for "Car #41" (i.e., p mod 100 = 41).
 
All the results are original by Mike Keith*.


      Q1. Would someone verify the 39 results obtained by M. Keith?

Q2. At what prime does Car 41 finally take the lead (if any)? The answer is known to be greater than 10^13. [Keith]

* Mr. Keith through G.L.H. sent the following image, that I share with all of you, if needed.


From Jan 17-23, 2026, contributions came from Jeff Heleen, Carlos Rivera,

***

Jeff wrote:

Q1. I got the same results as M. Keith.
 
Q2. No result for Q2 yet... Still less than 5 x 10^13.

***

 Carlos wrote:

I beg your pardon, but I just want to report other perhaps interesting things.When appearing the 93th prime, the prime 487, all the 42 primes residue(100), or "cars" (this include the cars/residues 02 & 05) has emerged at least one time. At this very moment, the first position in this race is shared by the 3 cars 31, 67 & 79 which have emerged four times, at the primes 31, 131, 331, 431; 67, 167, 367, 467; 79, 179, 379, 479; respectively. Of course the car 31 took the leadership at the prime 431, according to the figures provided by M. Keith above.

At this very moment in the second position (emerging only 3 times) are the 11 cars 7, 11, 13, 37, 73, 83, 97, 39, 49, 57 & 63.

In the third position (emerging only 2 times) are the 20 cars 3, 17, 19, 23, 29, 41, 43, 47, 53, 59, 61, 71, 89, 1, 9, 27, 51, 81, 93 & 33.

And in the last or fourth position (emerging only 1 time) are the 8 cars 2, 5, 91, 99, 69,77, 21 & 87.For a total of 3 + 11 + 20 + 8 = 42 cars

In my lite search the car 41 was at the third of fourth positions but never has been a leader, as we knew already. (I hope this is correct because I made the counting "by eye" or "by hand" in an Excel worksheet and a on a desktop calculator... Sorry in advance).

All this little search was made because I would like to know which is the relative position of the car 41 at the end of your respective searches? This little portion of additional information could tell us what are the possibilities of the car 41 to be the leader in the future... if this sometimes happens...

Currently I'm not programming anymore for different reasons; so if some of you can provide this little part of information I will include it here.

As a matter of fact I have asked by email this same question to Jeff Heleen during the week, because I knew that his countings were in process. So perhaps at the end of his search we will have some very good or very bad news... who knows?...

***

On Jan 29, 2026, Jeff Heleen sent the following results, related to my question: which is the relative position of the car 41 at the end of your respective searches?

His search may be summarized this way: at the height of the prime 95661860699  the car 41 is in the position No. 27th down the leader whici is the Car 47, while the Car 43 that according to the Keith's result was the last leader in this race, is now in the 12th position.

According to these results there are few chances at this very moment that the Car 41 could be a leader, unless that in the future his position gets better and better.

Heleen sent his results this way:
"Here's my last update. The numbers check out.
p = 95661860699 (prime)
counts = 
[98663579, 98666172, 98664121, 98667439, 98664890, 98666108, 98662245, 98664717, 98667968, 98665501, 98665309, 98662718, 98668280, 98666083, 98667696, 98665807, 98664893, 98666458, 98671663, 98664216, 98669442, 98666188, 98667231, 98666803, 98663557, 98664292, 98667839, 98664348, 98666519, 98663207, 98665846, 98665230, 98665095, 98667444, 98665792, 98666174, 98663629, 98665053, 98664411, 98665827]
 
Now someone with faster code can take over. Enjoy."

Here are the Heleen's results as a Table, constructed in Excel by me (CR). In color I have written the counts for the cars 41, 43 & 47.

Total Counts: 3946629790
P mod 100 Count of  Res Position
47 98671663 1
51 98669442 2
31 98668280 3
21 98667968 4
67 98667839 5
37 98667696 6
83 98667444 7
9 98667439 8
57 98667231 9
59 98666803 10
71 98666519 11
43 98666458 12
53 98666188 13
89 98666174 14
3 98666172 15
13 98666108 16
33 98666083 17
77 98665846 18
99 98665827 19
39 98665807 20
87 98665792 21
23 98665501 22
27 98665309 23
79 98665230 24
81 98665095 25
93 98665053 26
41 98664893 27
11 98664890 28
19 98664717 29
97 98664411 30
69 98664348 31
63 98664292 32
49 98664216 33
7 98664121 34
91 98663629 35
1 98663579 36
61 98663557 37
73 98663207 38
29 98662718 39
17 98662245 40

 

 

Records   |  Conjectures  |  Problems  |  Puzzles