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]


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?
3. 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?




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