Here are some observations that may be of interest. The following list of numbers are the primes from 0 through to 71 in sequence (including 1). If you add 1 to the square of the number you will make a total of 30030. So what? You may ask and rightly so! Except that 30030 is 13p# (1x2x3x5x7x11x13) which is odd, isn't it?
| 0 --> 1 |
| 1 --> 2 |
| 2--> 5 |
| 3--> 10 |
| 5--> 26 |
| 7--> 50 |
| 11--> 122 |
| 13--> 170 |
| 17--> 290 |
| 19--> 362 |
| 23--> 530 |
| 29--> 842 |
| 31--> 962 |
| 37--> 1370 |
| 41--> 1682 |
| 43--> 1850 |
| 47--> 2210 |
| 53--> 2810 |
| 59--> 3482 |
| 61--> 3722 |
| 67--> 4490 |
| 71--> 5042 |
|
| 30030 |
Also
11^2 + 13^2 + 17^2 + 19^2 +23^2 + 29^2 =
2310 (11p#)Here is another contiguous prime sequence - this is noteable as 199 is the first prime south side of 210 7(p#). You might expect this result if the primes were centred about 105 (p(7#)/2 but clearly they are not in this case.
199+211+223+227+229+233+239+241+251+257 =
2310 (11p#)also ( and this, I agree, is slightly more tenuous) but if you sum all the primes between 1 and 59 in the manner shown below
(1^2 + 2^2) + (2^2+3^2) + (3^2 + 5^2) + ....(47^2+53^2) + (53^2+59^2) =
30032(p#13 + 2)
Just observations that may help someone along the way .
BP 19/11/2009