is 2^n + 3^m a prime
 

I have written a C# code to find all the values of 2^m + 3^n and interestingly half of the first 76 primes were of the form 2^m + 3^n I did further research and realized that I should exclude all the combinations of m and n where the last digit would be a 5 for the resulting sum of 2^n and 3^m. specifically the last digit would be a 5 for the combinations where m is of form 4p and n is of form 4p+2 and (4p+1,4p+1) , (4p+2,4p+1) and (4p+3,4p+3). That is 4 combinations out of 16 possible combinations for the values of m and n. which leaves us 12 combinations possible for calculating the value of 2^m + 3^n which have a high probability to be a prime number.
I have also contemplated the possibility for the use of ternary number system to see if a relation exists for ternary numbers and decimal prime numbers, in vain. Please give me your thoughts on this

[QUOTE=blistervol;308271]half of the first 76 primes were of the form 2^m + 3^n[/QUOTE]

I don't know what you mean here. The 76th prime is 383.
26 of the first 76 primes can be written on the form 2^m + 3^n:
2, 3, 5, 7, 11, 13, 17, 19, 29, 31, 41, 43, 59, 67, 73, 83, 89, 97, 113, 131, 137, 251, 257, 283, 307, 337

[QUOTE=blistervol;308271]have a high probability to be a prime number[/QUOTE]

It doesn't look higher than similar random numbers.
A random number x has chance around 1/log(x) of being prime, where log is the natural logarithm.
Let's only consider positive m and n for simplicity. Then 2^m+3^n is never divisible by 2 or 3. You additionally exclude cases where it's divisible by 5.
If a random number x is not divisible by 2, 3 or 5 then the chance of primality increases by a factor 2/1 * 3/2 * 5/4 = 3.75, so it becomes 3.75/log(x).

For m and n in [1..1000] PARI/GP counts 6321 prp's of form x = 2^m+3^n. If we estimate that each of the numbers not divisible by 5 has chance 3.75/log(x) of primality then the expected number of primes is 6269. That's within 1% of the actual number.

[url]http://oeis.org/A004051[/url] has a link to the first 1000 primes of your form.

Here are some gigantic prp's:
[URL="http://www.primenumbers.net/prptop/searchform.php?form=3^m%2B2&action=Search"]3^m+2[/URL]
[URL="http://www.primenumbers.net/prptop/searchform.php?form=3^m%2B2^n&action=Search"]3^m+2^n[/URL] (currently they all have n = m+/-1 but that's just because somebody chose to search that form)
[URL="http://www.primenumbers.net/prptop/searchform.php?form=2^n%2B3&action=Search"]2^n+3[/URL]

Thank you for pointing me to that website.
 

[QUOTE=Jens K Andersen;308323]
Here are some gigantic prp's:
[URL="http://www.primenumbers.net/prptop/searchform.php?form=3^m%2B2&action=Search"]3^m+2[/URL]
[URL="http://www.primenumbers.net/prptop/searchform.php?form=3^m%2B2^n&action=Search"]3^m+2^n[/URL] (currently they all have n = m+/-1 but that's just because somebody chose to search that form)
[URL="http://www.primenumbers.net/prptop/searchform.php?form=2^n%2B3&action=Search"]2^n+3[/URL][/QUOTE]

I will be using my code to try and find more PRPs with 2^m + 3^n

Update
 

I Realized I was such a noob for thinking C# is my answer for calculating all the primes of that PRP. It has a really short range for storing variables. I am looking for more softwares that can crunch really big numbers. Also, Someone suggested using a custom data type in C#. Any suggestions would be awesome. :) thank you

[URL="http://pari.math.u-bordeaux.fr/"]Pari/gp[/URL] is great for prototyping/hypothesis testing.

[QUOTE=blistervol;308452]I Realized I was such a noob for thinking C# is my answer for calculating all the primes of that PRP. It has a really short range for storing variables. I am looking for more softwares that can crunch really big numbers. Also, Someone suggested using a custom data type in C#. Any suggestions would be awesome. :) thank you[/QUOTE]
As long as you are using a relatively recent version of c#(3.0+ i think) you can add as a reference system.numerics which adds a number called a BigInteger.