Sums of all Squares - Page 3

CRGreathouse · 2010-03-30, 17:27

Quote:

Originally Posted by bsquared

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 ;)

Maybe you can submit a sequence of every millionth term or something. Not exactly classy, but it has precedent (A080128, say).

76304519151822049179, 671924965564646162227, 2393465488665494654963, 5889405149040404480379, 11834774513923727795971, 20925456417823033330259, ...

bsquared · 2010-03-30, 19:00

Quote:

Originally Posted by CRGreathouse

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).

I just realized my code to do the sum of prime squares routines was hugely inefficient, in that I was using YAFU's built in arbitrary precision functions to do the squaring and summing. In reality we only need fixed precision of, say, 3 64 bit limbs (192 bits should be plenty to represent the sum). Implementing this made my sum of prime squares routine about 35 times faster.

I also store the highest power of 2 dividing the power of 10 modulus, which makes for a very quick pre-test of divisibility by 10 (logical AND followed by a predictable branch) and makes full precision divisions *extremely* rare. Doing things this way is actually faster than using pure modular arithmetic, since we almost never have to perform a division.

As a side benefit, it's easy to build tables of prime sums, or prime square sums, and we also don't have to restart a sum to test for a new power of 10 modulus.

bsquared · 2010-03-30, 19:02

Quote:

Originally Posted by CRGreathouse

Not exactly classy ...

Right up my alley, then

bsquared · 2010-03-30, 19:43

I must have some sort of tinkerers disease... can't leave well enough alone.

This disease was causing me to be offended by how long it was taking to compute the primes in a range of 1e9. So I tinkered... and doubled the speed

before:

Code:

found 40609038 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

after:

Code:

found 40609038 primes in range 49000000000 to 50000000000 in elapsed time = 2.8866
**** 49460594569 is 0 mod 10000000000 ****
sum of squares complete in elapsed time = 0.1639, sum is 1714863031171407826702942323341

which of course is completely useless, but now I feel better.

davar55 · 2010-03-30, 20:00

This is fine work by all of you. If you wish to submit the sequence to
oeis, please go ahead. I couldn't do justice to the calculations, which
I'm really impressed by. Joint discovery (attribution) is fine.

CRGreathouse · 2010-03-30, 20:14

Quote:

Originally Posted by davar55

I couldn't do justice to the calculations, which
I'm really impressed by.

I'll echo this. I just used a one-line Pari script to discover mine.

bsquared · 2010-04-02, 14:41

Now in OEIS: A174862

davar55 · 2010-04-06, 20:28

With all the work done on the OP, it shouldn't be too hard
to generalize the problem a bit.

I think cubes.

2^3 + 3^3 + 5^3 + ... + p^3 = 10^mK

What is the smallest prime p such that
the sum of cubes of all primes up to p
is a multiple of 10 (or 100 or 1000 or 10000 or ...).

I'm also curious about how these (squares and cubes) results
compare to first powers (sum of primes themselves).

Since these series depend on the properties of a number
in base ten, they could be considered recreational --
interesting but not necessarily useful. Still, perhaps the
sequence of sequences can someday be used to derive some
important number theoretic fact. That's one of the purposes
of the oeis.

petrw1 · 2010-04-06, 22:33

Quote:

Originally Posted by davar55

I'm also curious about how these (squares and cubes) results compare to first powers (sum of primes themselves).

A start on the first powers for the first 6,000,000 primes - p<=104,395,301

5 10
23 100
35677 63731000
106853 515530000
632501 15570900000

bsquared · 2010-04-06, 22:36

Quote:

Originally Posted by petrw1

A start on the first powers for the first 1,000,000 primes - p<=15,485,863

5 10
23 100
35677 63731000
106853 515530000
632501 15570900000

See here

petrw1 · 2010-04-06, 22:44

Quote:

Originally Posted by bsquared

See here

Cool....my first 5 answers match.

Crap...I missed the next 2: Bug alert!

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2010-03-30, 20:00	#27
davar55 May 2004 New York City 2³·23² Posts	This is fine work by all of you. If you wish to submit the sequence to oeis, please go ahead. I couldn't do justice to the calculations, which I'm really impressed by. Joint discovery (attribution) is fine.

2010-04-02, 14:41	#29
bsquared "Ben" Feb 2007 3,371 Posts	Now in OEIS: A174862

2010-04-06, 20:28	#30
davar55 May 2004 New York City 1000010001000₂ Posts	With all the work done on the OP, it shouldn't be too hard to generalize the problem a bit. I think cubes. 2^3 + 3^3 + 5^3 + ... + p^3 = 10^mK What is the smallest prime p such that the sum of cubes of all primes up to p is a multiple of 10 (or 100 or 1000 or 10000 or ...). I'm also curious about how these (squares and cubes) results compare to first powers (sum of primes themselves). Since these series depend on the properties of a number in base ten, they could be considered recreational -- interesting but not necessarily useful. Still, perhaps the sequence of sequences can someday be used to derive some important number theoretic fact. That's one of the purposes of the oeis.