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?
Q
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

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?

***

 

 



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