Prime Numbers

78 Chapter 1 PRIMES!
Next, as for quadruplets, argue heuristically that {p, p +2,p+6,p+8}
should be an allowed pattern. The current largest known quadruplet with this
pattern has its four member primes of the “titanic” class, i.e., exceeding 1000
decimal digits [Forbes 1999].
Next, prove that there is just one prime sextuplet with pattern: {p, p +
2,p+6,p+8,p+12,p+14}. Then observe that there is a prime septuplet
with pattern {p, p +2,p+6,p+8,p+12,p+18,p+20}; namelyforp = 11.
Find a different septuplet having this same pattern.
To our knowledge the largest septuplet known with the above specific
pattern was found in 1997 by Atkin, with first term
p = 4269551436942131978484635747263286365530029980299077\
1.86. Study the Smarandache–Wellin numbers, being
wn =(p1)(p2) ···(pn),
59380111141003679237691.
by which notation we mean that wn is constructed in decimal by simple
concatenation of the digits of the consecutive primes. For example, the first
few wn are 2, 23, 235, 2357, 235711, ....
First, prove the known result that infinitely many of the wn are composite.
(Hint: Use the fact established by Littlewood, that pi(x;3, 1) − pi(x;3, 2) is
unbounded in either (±) direction .) Then, find an asymptotic estimate (it
can be heuristic, unproven) for the number of Smarandache–Wellin primes
not exceeding x.
Incidentally the first “nonsmall” example of a Smarandache–Wellin prime
is
w128 = 23571113171923 ...719 .
How many decimal digits does w128 have? Incidentally, large as this example
is, yet larger such primes (at least, probable primes!) are known [Wellin 1998],
[Weisstein 2005].
1.87. Show the easy result that if k primes each larger than k lie in
arithmetic progression, then the common difference d is divisible by every
prime not exceeding k. Find a long arithmetic progression of primes. Note
that k = 22 was the 1995 record [Pritchard et al. 1995], but recently in 2004
[Frind et al. 2004] found that
56211383760397 + k · 44546738095860
is prime for each integer k ∈ [0, 22], so the new record is 23 primes. Factor
the above difference d = 44546738095860 to verify the divisibility criterion.
Find some number j of consecutive primes in arithmetic progression. The
current record is j = 10, found by M. Toplic [Dubner et al. 1998]. The
progression is {P + 210m : m =0,...,9}, with the first member being
P = 10099697246971424763778665558796984032950932468919004\
1803603417758904341703348882159067229719.