Problems & Puzzles:
Puzzles
Puzzle 234.
2003, again
This year (2003) has been outstanding
out there for several crucial situations. Arithmetically 2003 is also
outstanding ^{(*)}. 2003 is these kind of primes such that:
 2003, prime
 2x003+1= 7, prime
 20x03+1=61, prime
 200x3+1=601, prime
This is not the first time it happens
and not the least. But it seems that there are only a finite number of
primes with the shown property.
Q1. Can you find the largest prime of
these?
_____
^{(*)
}In one of the
Prime Curios! pages I
reported time
ago the following curio:
200*3 ± 1, 20*03 ± 1, and 2*003 ± 1 are three sets of
Twin prime
pairs. Note that 2003 is the largest prime for which this happens. [Rivera]
Solution:
Adam Stinchcombe and Faride
Firoozbakht found the same prime as probably the largest one as 2003:
6648647
Faride wrote:
The largest prime less than prime(155*10^6) of
these, is 6648647. It is interesting that, 6648647 is also the largest odd
number of these (that I found). I found more than 200 numbers of these
(greater than 6648647)which all of them are even numbers.
Q2.
Can you find the largest numbers of these?
Q3. Is there exist odd
number greater than 6648647 of these?
***
Giovanni Resta responded to
the Faride's Q3:
Yep!. I stopped my search at 10,000,000,000 and
these are the largest odd numbers
28222061
60404485
60882665
642846005
8640028405
while the largest even number found was 9895016332.
***
Faride Firoozbakht added:
The largest number for Q2 that was found By Giovanni
Resta is "n=9895016332". I found "m=12250053526",which is greater than n.
Q2. Can you find the largest numbers of these?
***
