SQLServerCentral / Discuss Content Posted by Kathi Kellenberger / Article Discussions / Article Discussions by Author / Finding Primes / Latest Posts

RE: Finding Primes

Mitch Wheat — Mon, 31 Dec 2007 22:32:13 GMT

Can I suggest you re-read the book: "...of the story is a 15 year old autistic boy" He is not autistic. He has aspergers syndrome. Whilst there are some similarities, they are world's apart.

RE: Finding Primes

Michael Valentine Jones — Wed, 31 Jan 2007 16:37:00 GMT

Ryan, I modified your script to use INT temp tables, instead of BIGINT, and got almost an 18% reduction in runtime. The following is the result of finding all primes below 100 million.

Using BIGINT for the data types doesn't seem to be needed unless you are going over 2 billion in your prime number search.

Test with INT temp tablesCheckpoint A: 250 msCheckpoint B: 230573 msFinish: 652583 msNumber of primes Max prime   Average of primes    ---------------- ----------- -------------------- 5761455          99999989    48461680

Test with BIGINT temp tablesCheckpoint A: 250 msCheckpoint B: 280590 msFinish: 791850 msNumber of primes Max prime            Average of primes    ---------------- -------------------- -------------------- 5761455          99999989             48461680

RE: Finding Primes

Jesper-244176 — Wed, 31 Jan 2007 01:44:00 GMT

I have tested my code against Ryan's code on SQL Server 2000 and it seems to be equally fast (with the same output).If the Numbers table is preconstructed, I guess my code is faster (because I need a larger Numbers table). On the other hand, if I am forced to use bigints (as Ryan does), my code willbe a bit slower.

I have used the following in my test:

/*************** MY CODE ******************/

set nocount ongo

declare @time datetimeselect @time = getdate()

DECLARE @MaxNumber INTSET @MaxNumber = 5000000

SET ROWCOUNT @MaxNumber

select identity(int, 1, 1) as Number into #Numbersfrom syscomments c1 cross join syscomments c2 cross join syscomments c3

SET ROWCOUNT 0

alter table #Numbers add constraint PK_Numbers primary key clustered (Number)

create table #Primes(prime int primary key)

declare @i int, @limit intset @i = 1while @i*@i < @MaxNumberbegin set @limit = (@i+1)*(@i+1) if (@i+1)*(@i+1) > @MaxNumber set @limit = @MaxNumber + 1

insert into #Primes select n.Number from #Numbers n where n.Number < @limit and n.Number > @i*@i and not exists (select * from #Primes p where p.prime < @i + 1 and n.Number % p.prime = 0)

set @i = @i + 1end

--select * from #Primes

PRINT 'Finish: ' + CAST(DATEDIFF(ms, @Time, GETDATE()) AS VARCHAR(10)) + ' ms'SELECT COUNT(*) AS 'Number of primes', MAX(prime) AS 'Max prime' FROM #Primes

drop table #Primesgo

drop table #Numbersgo

/******** RYAN'S CODE************************/

--InputsDECLARE @MaxNumber INTSET @MaxNumber = 5000000

--PreparationSET NOCOUNT ONDECLARE @Time DATETIMESELECT @Time = GETDATE()

--Preparation - Numbers tableDECLARE @SqrtMaxNumber INTSET @SqrtMaxNumber = SQRT(@MaxNumber)SET ROWCOUNT @SqrtMaxNumberCREATE TABLE #Numbers (i BIGINT IDENTITY(1, 1) PRIMARY KEY CLUSTERED, j BIGINT, x bit)INSERT INTO #Numbers (x) SELECT NULL FROM syscomments c1 CROSS JOIN syscomments c2SET ROWCOUNT 0UPDATE #Numbers SET j = i*iPRINT 'Checkpoint A: ' + CAST(DATEDIFF(ms, @Time, GETDATE()) AS VARCHAR(10)) + ' ms'

--Preparation - Put candidate primes into a Primes table-- (integers which have an odd number of representations by certain quadratic forms)CREATE TABLE #Primes (i BIGINT PRIMARY KEY CLUSTERED)INSERT #PrimesSELECT 2 UNION ALL SELECT 3 UNION ALLSELECT k FROM ( SELECT k FROM (SELECT 4 * a.j + b.j AS k FROM #Numbers a, #Numbers b) c WHERE k <= @MaxNumber AND k % 12 IN (1, 5) UNION ALL SELECT k from (select 3 * a.j + b.j AS k FROM #Numbers a, #Numbers b) c WHERE k <= @MaxNumber AND k % 12 = 7 UNION ALL SELECT k from (select 3 * a.j - b.j AS k FROM #Numbers a INNER JOIN #Numbers b ON a.i > b.i) c WHERE k <= @MaxNumber AND k % 12 = 11) d GROUP BY k HAVING COUNT(*) IN (1, 3)PRINT 'Checkpoint B: ' + CAST(DATEDIFF(ms, @Time, GETDATE()) AS VARCHAR(10)) + ' ms'

--Calculation - Eliminate composites by sievingDECLARE @i BIGINTSET @i = 5WHILE @i * @i < @MaxNumberBEGIN DELETE #Primes WHERE i > @i and i % @i = 0 SELECT @i = min(i) FROM #Primes WHERE i > @i-- PRINT @iEND

--Show resultsPRINT 'Finish: ' + CAST(DATEDIFF(ms, @Time, GETDATE()) AS VARCHAR(10)) + ' ms'SELECT COUNT(*) AS 'Number of primes', MAX(i) AS 'Max prime' FROM #Primes--SELECT * FROM #Primes ORDER BY i

--Tidy upDROP TABLE #PrimesDROP TABLE #Numbers

RE: Finding Primes

Michael Valentine Jones — Tue, 30 Jan 2007 22:31:00 GMT

Good stuff Ryan!

Did you compare the runtime against any of the other code posted on this thread?

RE: Finding Primes

RyanRandall — Tue, 30 Jan 2007 06:50:00 GMT

Here's a selection of results when I run the above code on a server here.

39 seconds to find all primes below 5 million, and 18 minutes to find all primes below 100 million.

DECLARE @Results TABLE (MaxNumber INT, CheckpointA INT, CheckpointB INT, Finish INT, NumberOfPrimes INT,

MaxPrime INT, AverageOfPrimes INT,

Seconds AS CAST(Finish AS DECIMAL) / 1000, Minutes AS CAST(Finish AS DECIMAL) / 60000,

Time AS CAST(Finish / 60000 AS VARCHAR(5)) + ':' + RIGHT('0' + CAST(Finish / 1000 % 60 AS VARCHAR(2)), 2))

INSERT @Results

SELECT 1000, 60, 93, 106, 168, 997, 453

UNION ALL SELECT 5000, 76, 123, 140, 669, 4999, 2314

UNION ALL SELECT 10000, 76, 156, 186, 1229, 9973, 4667

UNION ALL SELECT 50000, 76, 280, 360, 5133, 49999, 23575

UNION ALL SELECT 100000, 76, 466, 656, 9592, 99991, 47372

UNION ALL SELECT 500000, 76, 1250, 2576, 41538, 499979, 238678

UNION ALL SELECT 1000000, 93, 2203, 5516, 78498, 999983, 478361

UNION ALL SELECT 5000000, 110, 10156, 39280, 348513, 4999999, 2406213

UNION ALL SELECT 10000000, 93, 20030, 93403, 664579, 9999991, 4820081

UNION ALL SELECT 50000000, 123, 208930, 843080, 3001134, 49999991, 24197369

UNION ALL SELECT 100000000, 140, 466970, 1089933, 5761455, 99999989, 48461680

SELECT * FROM @Results

/* Results

MaxNumber CheckpointA CheckpointB Finish NumberOfPrimes MaxPrime AverageOfPrimes Seconds Minutes Time

--------- ----------- ----------- ------- -------------- -------- --------------- ----------- ---------- -----

1000 60 93 106 168 997 453 0.106000 0.001766 0:00

5000 76 123 140 669 4999 2314 0.140000 0.002333 0:00

10000 76 156 186 1229 9973 4667 0.186000 0.003100 0:00

50000 76 280 360 5133 49999 23575 0.360000 0.006000 0:00

100000 76 466 656 9592 99991 47372 0.656000 0.010933 0:00

500000 76 1250 2576 41538 499979 238678 2.576000 0.042933 0:02

1000000 93 2203 5516 78498 999983 478361 5.516000 0.091933 0:05

5000000 110 10156 39280 348513 4999999 2406213 39.280000 0.654666 0:39

10000000 93 20030 93403 664579 9999991 4820081 93.403000 1.556716 1:33

50000000 123 208930 843080 3001134 49999991 24197369 843.080000 14.051333 14:03

100000000 140 466970 1089933 5761455 99999989 48461680 1089.933000 18.165550 18:09

RE: Finding Primes

RyanRandall — Tue, 30 Jan 2007 06:40:00 GMT

Hi Jeff,

Here's some code so you can run it yourself. I'll add some run results in the next post.

--InputsDECLARE @MaxNumber INTSET @MaxNumber = 5000000

--PreparationSET NOCOUNT ONDECLARE @Time DATETIMESELECT @Time = GETDATE()

--Preparation - Numbers tableDECLARE @SqrtMaxNumber INTSET @SqrtMaxNumber = SQRT(@MaxNumber)SET ROWCOUNT @SqrtMaxNumberCREATE TABLE #Numbers (i BIGINT IDENTITY(1, 1) PRIMARY KEY CLUSTERED, j BIGINT, x bit)INSERT INTO #Numbers (x) SELECT NULL FROM syscomments c1, syscomments c2, syscomments c3, syscomments c4SET ROWCOUNT 0UPDATE #Numbers SET j = i*iPRINT 'Checkpoint A: ' + CAST(DATEDIFF(ms, @Time, GETDATE()) AS VARCHAR(10)) + ' ms'

--Show resultsPRINT 'Finish: ' + CAST(DATEDIFF(ms, @Time, GETDATE()) AS VARCHAR(10)) + ' ms'SELECT COUNT(*) AS 'Number of primes', MAX(i) AS 'Max prime', AVG(i) AS 'Average of primes' FROM #PrimesSELECT * FROM #Primes ORDER BY i

--Tidy upDROP TABLE #PrimesDROP TABLE #Numbers

RE: Finding Primes

Jesper-244176 — Tue, 30 Jan 2007 02:59:00 GMT

My suggestion, on our SQL Server 2000 installation it finds all primes less than 5000000 in 54 seconds. But of course it depends on the hardware...

set nocount ongo

select top 5000000 identity(int, 1, 1) as Number into #Numbersfrom syscomments c1 cross join syscomments c2go

alter table #Numbers add constraint PK_Numbers primary key clustered (Number)go

create table #Primes(prime int primary key)go

declare @time datetimeselect @time = getdate()

declare @i intset @i = 1while @i*@i < 5000000begin insert into #Primes select n.Number from #Numbers n where n.Number < (@i+1)*(@i+1) and n.Number > @i*@i and not exists ( select * from #Primes p where p.prime < @i + 1 and n.Number % p.prime = 0

set @i = @i + 1end

--select * from #Primes

select datediff(ms, @time, getdate())

drop table #Primesgo

drop table #Numbersgo

RE: Finding Primes

Jeff Moden — Mon, 29 Jan 2007 20:31:00 GMT

Hi Ryan,

What did you come up with for the largest prime number found and how many prime numbers? I wanna make sure I'm doing it right...

RE: Finding Primes

Mitch Wheat-379974 — Mon, 29 Jan 2007 17:32:00 GMT

Kathi

In the book you mentioned, Chris has Aspbergers Syndrome (rather than being autistic). There are many similarities, but people with Aspbergers for the most part can lead relatively 'normal' lives. There are varying degrees of both of course, and it's sometimes hard to make a clear distinction where one ends and the other begins.

Regards,

MItch

RE: Finding Primes

GregLyon — Mon, 29 Jan 2007 12:37:00 GMT

Fun stuff.I tried to do something similar in VB several years ago...the trick was that I was trying to solve for primes bigger than any of the data types. I ran out of time before I solved it...but it sure was a fun exercise!

RE: Finding Primes

Rob Farley — Mon, 29 Jan 2007 10:58:00 GMT

More on primes at:http://blogs.technet.com/wardpond/archive/2006/09/23/458344.aspx http://blogs.technet.com/wardpond/archive/2006/09/23/458580.aspx http://sqlblog.com/blogs/hugo_kornelis/archive/2006/09/23/Prime_numbers.aspx http://sqlblog.com/blogs/hugo_kornelis/archive/2006/09/23/Prime_numbers.aspx#commentsAnd of course, at my _old_ blog at:http://robfarley.blogspot.com/2006/09/primes.html http://robfarley.blogspot.com/2006/09/more-on-primes.htmlAll lots of fun...

RE: Finding Primes

Kathi Kellenberger — Mon, 29 Jan 2007 08:59:00 GMT

Good idea.

RE: Finding Primes

Jason-262847 — Mon, 29 Jan 2007 08:49:00 GMT

By checking the square root outside of the select statement, you will save the processor a number of extra calculations. I modified it to be this:

SELECT @NextIntSqrt = SQRT(@NextInt) IF NOT EXISTS (SELECT Prime FROM Primes WHERE @NextIntSqrt >= Prime AND @NextInt % Prime = 0)

Cuts down the calculation time significantly. I also adjusted the order of the filters in the WHERE clause.

RE: Finding Primes

Kathi Kellenberger — Mon, 29 Jan 2007 03:26:00 GMT

Cool. Looks like there are a lot of ways to solve this puzzle.

RE: Finding Primes

RyanRandall — Mon, 29 Jan 2007 03:07:00 GMT

Finding primes is something I worked on about 6 months ago (for fun, of course ). Here's a link to the work...http://www.sqlteam.com/forums/topic.asp?TOPIC_ID=69646It uses a sieve of Atkin approach, and in my tests back then, the function could find all primes below 100,000 in 0.5 seconds.

Finding Primes

Kathi Kellenberger — Wed, 27 Dec 2006 11:43:00 GMT

Comments posted here are about the content posted at http://www.sqlservercentral.com/columnists/kKellenberger/2782.asp