Problems & Puzzles: Puzzles
Puzzle 257. Primes and sibling numbers
The prime 727 is the first prime such that its square 528529 is a sibling number of the type a.(a+1) where a=528 and here the dot "." means concatenation.
So, in this puzzle we are interested in primes p such that p2 = a.(a+1)
Q1. Find the next 3 primes of this type [p2 = a.(a+1)]
Q2. Find a titanic prime of this type [p2 = a.(a+1)].
Solutions to Q1 & Q2 were sent by J. C. Rosa and Patrick de Geest, respectively.
Q1. J. C. Rosa has found the first eight primes of the asked type:
BTW, his solution 37380894211was already published in the Plate 152 of the well known Patrick de Geest's site, World of Numbers.
This is the JCR's method:
Q2. Patrick de Geest sent the following solutions:
Also Jean Claude Rosa made an important contribution to the topic.
In the subsection " Near-tautonymic numbers of the form (T)_(T + 1) as SQUARES " one can find the first two primes as asked in question Q1 :
[ 727 ]^2 = 528_529
Unfortunately the list ends before the third and fourth prime showed up. No doubt Jean-Claude Rosa will give the next few terms.
What does show up in the list is a pattern starting from the given example 727 :
[ (6n)7(3n)2(6n)7 ]^2 = (4n)5(3n)2(8n)8_(4n)5(3n)2(8n)9
(6n) means '6' repeated n times
By putting the rootnumber in an ABC2 file for PFGW I was able to find some n values yielding probable primes. I hope this suffices as a solution to question Q2 of your puzzle. The puzzle doesn't ask for the smallest titanic solution only "Find 'a' titanic prime...". primes for the following n values = 0, 16, 17, 33. probable primes for n = 2738, 3096
Note that the digit lengths of these last two '3-PRP' are respectively 8217 and 9291 and so are almost gigantic ones ! I shall try to find a genuine gigantic probable prime solution.
Phil Carmody found (March 2003) the earliest (supposedly) titanic prime of the asked type (only the first 10 initial and 10 ending digits shown, CR; the whole prime can be sent on request)
5107962136...9079192809, 1000 digits
See his method described in the Puzzle 258.