Hi,
I tried to figure out the probability for this several times but was unsuccessful, please figure it out if you can:
Statistical distribution of likely next last digits of primes
The way that is easiest to see it is for primorial 30 equation sets that produce all primes (except 2,3,5) with the formula y=mx+b
y = 30x + 1 y = 30x + 7 y = 30x + 11 y = 30x + 13 y = 30x + 17 y = 30x + 19 y = 30x + 23 y = 30x + 29
y = 30x + 1 (produces half of primes ending in 1) y = 30x + 7 (produces half of primes ending in 7) y = 30x + 11 (produces half of primes ending in 1) y = 30x + 13 (produces half of primes ending in 3) y = 30x + 17 (produces half of primes ending in 7) y = 30x + 19 (produces half of primes ending in 9) y = 30x + 23 (produces half of primes ending in 3) y = 30x + 29 (produces half of primes ending in 9)
Each equation produces half of all primes ending in a give digit 1,3,7,9
Here are the prime1, prime2 consecutive last digit occurrence counts for the 4 permutations of b values 1,11 that can have consecutive primes both ending in 1: (full table at bottom of page)
p1,p2,count
1,11,2159
11,1,547
11,11,266
1,1,241
The sum of the four counts is much lower, than for example primes p1,p2 where p1 ends in 1, and p2 ends in 7:
p1,p2,count
1,7,2849
11,17,2193
1,17,1087
11,7,421
Right there that shows for a given primorial and a number range you can see the likelihood of consecutive primes ending digits, however I didn't do a full statistical formula considering all primorials and any number range, tried and gave up for now :D
Please figure out a generalized formula for any number range to show the probability of a two consecutive last primes last digit occurrence.
cheers, Jamie
full table:
p1,p2,count
7,11,2945
11,13,2916
17,19,2914
29,1,2904
23,29,2893
13,17,2892
19,23,2887
1,7,2849
23,1,2209
11,17,2193
17,23,2181
13,19,2163
1,11,2159
19,29,2154
29,7,2153
7,13,2087
23,7,1824
7,17,1624
13,23,1590
1,13,1585
19,1,1561
11,19,1551
17,29,1551
29,11,1526
29,13,1345
17,1,1278
13,29,1160
7,19,1138
11,23,1133
1,17,1087
19,7,1020
23,11,993
19,11,957
1,19,811
7,23,795
11,29,783
29,17,774
13,1,737
17,7,724
23,13,720
1,23,665
7,29,633
13,7,560
11,1,547
23,17,543
29,19,519
19,13,517
17,11,516
13,11,448
19,17,431
23,19,431
11,7,421
1,29,410
17,13,380
29,23,363
7,1,330
13,13,274
11,11,266
17,17,265
19,19,261
7,7,260
1,1,241
23,23,226
29,29,221