## Wednesday, April 13, 2011

### Square root of summed prime numbers

Let's take the prime numbers below 500

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499

The aim: to identify all sets of N consecutive primes such that the square root of their sum is prime.

N = 2 to 9.

1. 13 17 19 Sum 49
37 41 43 Sum 121
277 281 283 Sum 841
313 317 331 Sum 961
181 191 193 197 199 Sum 961
317 331 337 347 349 Sum 1681
13 17 19 23 29 31 37 Sum 169
293 307 311 313 317 331 337 Sum 2209
73 79 83 89 97 101 103 107 109 Sum 841
227 229 233 239 241 251 257 263 269 Sum 2209

Paul C.

2. Good. Your upper limit was 269.
49 = 7^2, 121 = 11^2,
841 = 29^2, 961 = 31^2
1681 = 41^2, 2209 = 47^2
(7,11,29,31,41,47)

3. We could extend the range of primes to say 100000 and N=3 we find

13 17 19 49 7
37 41 43 121 11
277 281 283 841 29
313 317 331 961 31
613 617 619 1849 43
7591 7603 7607 22801 151
8209 8219 8221 24649 157
12157 12161 12163 36481 191
23053 23057 23059 69169 263
32233 32237 32251 96721 311
42953 42961 42967 128881 359
44887 44893 44909 134689 367

and a N=19

87323 87337 87359 87383 87403 87407 87421 87427 87433 87443 87473 87481 87491 87509 87511 87517 87523 87539 87541 Sum 1661521 1289

We observe that there are no solutions when N is even because the sum of an even number of odd numbers is always even and the squareroot of an even number is always even and thus cannot be a prime.

paul