>Would you by chance know who was the first mathematician to establish
>that, if N primes are in arithmetical progression, the difference between
>two consecutive terms is a multiple of each prime smaller than N ?
>Or if this result has been attributed ñ fairly or not ñ to some mathematician?
Theorem of M. Cantor:
If n (n>5) primes are in arithmetical progression, then the progr.'s common
difference is divisible by every prime <=n
If I remember correctly, I have read it in:
W. Sierpinski: 250 Problems in Elementary Number Theory. New York, 1970
Example: n=10, a_1 = 199, dif. = 210
Seredinsky's arith. prog.: n = 13; a_1 = 4943; dif. = 60060 = 2.2.3.5.7.11.13
S. C. Root's (1969) : n = 16; a_1 = 2,236,133,941; dif.= 223,092,870
For 22 primes in arith. progression
(11410337850553 + 4609098694200 i, 0 <= i < 22)
read the posting at:
http://listserv.nodak.edu/scripts/wa.exe?A2=ind9303&L=nmbrthry&P=R258
For 7 *consecutive* primes in arith, progression read the postings at:
http://listserv.nodak.edu/scripts/wa.exe?A2=ind9508&L=nmbrthry&P=R220
http://listserv.nodak.edu/scripts/wa.exe?A2=ind9803&L=nmbrthry&P=R225
For 8 *consecutive* p.:
http://listserv.nodak.edu/scripts/wa.exe?A2=ind9711&L=nmbrthry&P=R215
For 9 *consecutive* p.:
http://listserv.nodak.edu/scripts/wa.exe?A2=ind9801&L=nmbrthry&P=R829
For 10 *consecutive* p.:
http://listserv.nodak.edu/scripts/wa.exe?A2=ind9803&L=nmbrthry&P=R22
PS: An Analytic Number Theory Bibliography can be found at:
http://math.uwaterloo.ca/~shallit/bib/analytic.bib
This B. contains some items on primes on arith. progression you may try
for possible historical information.
It's part of Algorithmic Number Theory - Bibliography (by Eric Bach and
Jeffrey Shallit), located at:
http://math.uwaterloo.ca/~shallit/antbib.html
aph