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 1,1,241The 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 11,7,421Right 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