000RM.dvi

22.3 The prime number spiral beginning with 41 619

Appendix: Consecutive primes with consecutive prime digital sums

Charles Twigg asked, inCrux Math., Problem 228, for four consecutive
primes having digital sums that, in some order, are consecutive primes.
And then five.
The beginning of the prime number sequence provides an easy an-
swer: just consider the primes 2, 3, 5, 7, or 3, 5, 7, 11. Beyond these, the
first quadruple is 191, 193, 197, and 199, with digit sums 11, 13, 17, 19.
The five consecutive primes 311, 313, 317, 331, 337 all have prime
digital sums, though these are the same for 313 and 331.
The first sequence of five consecutive primes who digital sums form
another sequence of 5 consecutive primes is

(1291,13), (1297,19),(1301,5),(1303,7),(1307,11).

Twigg listed such quadruples and quintuples up to primes around 5
million (about 350000 primes). But within this range he had overlooked
the better records

(102251,11),(102253,13),(102259,19),(102293,17),(102299,23),(102301,7)

and

3511973 ,29 3511993,31 3511999,37 3512011, 13

3512051 ,17 3512053,19 3512057, 23

A little bit beyond these we find the best record for eight up to the first 1
million primes:

5919931 ,37 5919937,43 5919959,47 5919971, 41

5920003 ,19 5920043,23 5920049,29 5920069, 31

Within the same range, there are also 15 consecutive primes whose
digital sums are primes, though only with 5 different values:

2442113 ,17 2442133,19 2442151,19 2442173,23 2442179, 29

2442191 ,23 2442197,29 2442199,31 2442227,23 2442263, 23

2442287 ,29 2442289,31 2442311,17 2442353,23 2442359, 29

Another “long” chain of 9 consecutive primes with 5 different con-
secutive prime digital sums can be found among “small” primes:

14293 ,19 14303,11 14321,11 14323,13 14327, 17

14341 ,13 14347,19 14369,23 14387, 23