Problems & Puzzles:
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
In one of the
Prime Curios! pages I
ago the following curio:
200*3 ± 1, 20*03 ± 1, and 2*003 ± 1 are three sets of
pairs. Note that 2003 is the largest prime for which this happens. [Rivera]
Adam Stinchcombe and Faride
Firoozbakht found the same prime as probably the largest one as 2003:
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.
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
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?