update...
[CODE]found primes in range 30000000000 to 31000000000 in elapsed time = 5.4245 sum of squares complete in elapsed time = 7.1620, sum is 416903941002774697723222981803 found primes in range 31000000000 to 32000000000 in elapsed time = 5.5802 **** 31252968359 is 0 mod 1000000000 **** sum of squares complete in elapsed time = 7.1759, sum is 457955303775896882861615585442 found primes in range 32000000000 to 33000000000 in elapsed time = 5.5252 sum of squares complete in elapsed time = 7.1461, sum is 501598601070515778427418232428 [/CODE] and another update: [CODE]found primes in range 47000000000 to 48000000000 in elapsed time = 5.5268 sum of squares complete in elapsed time = 7.0336, sum is 1519756369296424391708040649758 found primes in range 48000000000 to 49000000000 in elapsed time = 5.4239 sum of squares complete in elapsed time = 7.0347, sum is 1615357580573805620690452754303 found primes in range 49000000000 to 50000000000 in elapsed time = 5.4835 **** 49460594569 is 0 mod 1410065408 **** sum of squares complete in elapsed time = 6.8852, sum is 1714863031171407826702942323341 found primes in range 50000000000 to 51000000000 in elapsed time = 5.4299 [/CODE] I'm stupidly using %u to print the modulus, but it is stored internally as a 64 bit integer... so it is only a printing error. 
Right on cue
we have this result:
[CODE]found primes in range 1915000000000 to 1916000000000 in elapsed time = 5.0965 **** 1915014433303 is 0 mod 1215752192 **** sum of squares complete in elapsed time = 5.0550, sum is 83903230112675776937166385335972895 [/CODE] So the sum of primes squared up to 1915014433303 is zero mod 100e9. I'm processing about 1 billion numbers per second, so assuming the trend of this sequence holds, to go to the next value at around 20 trillion would take a couple days. 
[quote=bsquared;210004]...I'm processing about 1 billion numbers per second...[/quote]
That should have been 1e9 every 10 sec, but the time estimate is still about right. It's probably silly, but a run to 20 trillion is ongoing... Here was the sum of all prime squares up to 1915014433303: 83775363722237720731978600000000000 I'm keeping a file with the sums every 1e9, in case anyone wants to extend the sequence after I get tired of it or for double checks. 
I see that the death knell of this thread has been sounded :smile:
Maybe the 12th member of the sequence is still of interest... [tex]\Sigma_{p = 2}^{4076200167673} p^2 = 786646994677132840800629000000000000[/tex] 
[QUOTE=bsquared;210051]I see that the death knell of this thread has been sounded :smile:
Maybe the 12th member of the sequence is still of interest... [tex]\Sigma_{p = 2}^{4076200167673} p^2 = 786646994677132840800629000000000000[/tex][/QUOTE] Are you going to submit this to the OEIS? 
I have no problem doing so... but I'm not the "discoverer" of this sequence. I'll defer to you or davar55 if you would rather take the credit.

I PM'd davar55.

Also, [url=http://www.research.att.com/~njas/sequences/A024450]this sequence[/url] could now be greatly extended.

[QUOTE=bsquared;210069]Also, [url=http://www.research.att.com/~njas/sequences/A024450]this sequence[/url] could now be greatly extended.[/QUOTE]
Sure. It's quite easy to extend, but tradition limits bfiles to 10,000 entries. If you'd like you can extend it to that (or I can), but I wouldn't go beyond. The current bfile has 5000. As it happens I never computed that sequence for these calculations  I used pure modular arithmetic. (If I used BCD I could have avoided this while keeping speed high...) 
[QUOTE=CRGreathouse;210070] Sure. It's quite easy to extend, but tradition limits bfiles to 10,000 entries. If you'd like you can extend it to that (or I can), but I wouldn't go beyond. The current bfile has 5000.
[/QUOTE] I guess I was thinking of a link to tables, or something. But that would require me to generate and host those tables. I'll save that for some day when I'm bored ;) [QUOTE=CRGreathouse;210070] As it happens I never computed that sequence for these calculations  I used pure modular arithmetic. (If I used BCD I could have avoided this while keeping speed high...) [/QUOTE] Yeah, that's definitely faster, but using pure modular arithmetic would require you to start the sum over for each new modulus, right? 
[QUOTE=bsquared;210073]Yeah, that's definitely faster, but using pure modular arithmetic would require you to start the sum over for each new modulus, right?[/QUOTE]
Right. You could do two modili at a time without much penalty, though, with appropriate lookup tables and bit operations. Note that you only need to compare (and hence reduce) every 8 primes, each term (other than the first) has index = 5 (mod 8). 
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