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Make a program that calculates the value of the 1,000th, the 10,000th and the 100,000th prime. If FB is used the 1,000,000th prime must be calculated too.
The first prime is 2, the second is 3 and so on...
The number of primes up to the integer x can be aproximated by x/(log(x)1) (QB's log)
The winner is who has the three primes right and faster.
References:
All you wanted to know about primes and never dared to ask http://primes.utm.edu/
The idea is to generate primes and count them up to the required count. There are a lot of optimizations possible in this generation and counting so I hope we will get some interesing sources.
We can open 3 categories according to the speed and memory limitations: Qbasic, Qb4.5 and FreeBASIC
EDITED: Added an additional request for FB, without it even an unoptimized souce takes less than 0.5 second.
EDIT2: For reference, the results are
Code: the 1000 th prime is 7919
the 10000 th prime is 104729
the 100000 th prime is 1299709
the 1000000 th prime is 15485863
of course your program must FIND these results!
Antoni
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err winer? :lol:
\__/)
(='.'=) Copy bunny into your signature to
(")_(") help him gain world domination.
Posts: 1,407
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Posts: 484
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Joined: Apr 2005
Heres my simple FB prime program, using trial divison. It basically combines the only two rules i know, that theres no need to look at even numbers (except 2), and that you only need to check if n is divisible by any prime numbers smaller than sqr(n)
It takes about 0.5 seconds to get the 100,000 prime, and about 12.5 to get the 1,000,000 on my P4 1.8Ghz
I've been reading the link Antoni posted and may try again once i learn some different methods.
Code: Dim Shared prime_list(1 To 1000000) As uInteger
Dim Shared prime_list_count As uInteger
Dim As uInteger prime_val, i, j, n
Dim As Integer is_prime
prime_list(1) = 2 ' Setup the first two values
prime_list(2) = 3
prime_list_count = 2
prime_val = 3
Do
prime_val += 2 ' Add 2, we don't need to look at even numbers
is_prime = 1
n = prime_val
i = 1
j = Int(sqr(n)) + 1 ' Only need to check if n is divisible by any prime smaller than sqr(n)
While prime_list(i) < j
If (n mod prime_list(i)) = 0 Then
is_prime = 0
Exit While
End If
i += 1
Wend
If is_prime Then
prime_list_count += 1
prime_list(prime_list_count) = prime_val
End If
Loop Until prime_list_count = 1000000
Print prime_list(1000)
Print prime_list(10000)
Print prime_list(100000)
Print prime_list(1000000)
EVEN MEN OF STEEL RUST.
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Great, we have one entry!
But sieves are faster...
Antoni
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Heres another one, a fairly standard sieve, using a bit array. I tried it just using a regular array which will take much more memory, and i though would be faster, but this bit based method won in the timings.
I also tried to change IsGood to a macro, but that actually slowed it down, I think thats P4 weirdness.
The only real optimization i did for this was the multiples of two, which because i used a bit array, i just iterated through, and masked out all multiple of two numbers.
This one runs in about 3.5 to 4 seconds, so 1/3 of the time of my first attempt, and i'm sure there are better methods yet...
Code: Const MaxPrime = 1000000 ' The max value for P
Const MaxVal = MaxPrime * 16 ' The max value for N (a bit of a cheat using * 16...)
Const Max32 = (MaxVal \ 32) + 1 ' The number of 32bit vars needed to store bit array
Dim Shared BitsArray(0 To Max32) As uInteger
#macro MarkGood(n)
Scope
Dim As uInteger p = (n) \ 32, o = (n) mod 32
BitsArray(p) = BITRESET(BitsArray(p), o)
End Scope
#endmacro
#macro MarkBad(n)
Scope
Dim As uInteger p = (n) \ 32, o = (n) mod 32
BitsArray(p) = BITSET(BitsArray(p), o)
End Scope
#endmacro
Function IsGood(ByVal n As uInteger) As Integer
Dim As uInteger p = n \ 32, o = n mod 32
Return NOT BIT(BitsArray(p), o)
End Function
Dim As uInteger i, n1, n2, count, mask = &H55555555
For i = 0 To Max32  1 ' Mark off all multiples of two quickly using a mask of 10...
BitsArray(i) = mask
Next i
MarkGood(2) ' Restore 2 as a prime
MarkBad(1) ' Make 1 not a prime
count = 1 ' start count offset at 1 to account for 2 being a prime
For n1 = 3 To MaxVal
If IsGood(n1) Then
count += 1
If count = 1000 Then Print n1
If count = 10000 Then Print n1
If count = 100000 Then Print n1
If count = 1000000 Then
Print n1
Exit For ' We've found the 1 millionth prime, we can quit
End If
For n2 = (n1 + n1) To MaxVal Step n1 ' work from n+n to max marking off multiples
MarkBad(n2)
Next n2
End If
Next n1
EVEN MEN OF STEEL RUST.
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This is an interesting source by Rich Geldreich anyone can find at ABC packets. It uses an original idea and it's probably the only way to do it in QBasic 1.1, because of the 160K memory limits.
It ran in 14 secondes in QB1.1 and in 2 seconds compiled in QB4.5
Code: 'Prime tally using a moving window version of the Erathostenes' Sieve
'Antoni Gual 10/2006 for the comtest at QBN. Qbasic1.1 version
'
'A true bit sieve would be faster, but the memory sizes in QB1.1
'require bold ideas. This one was created for QB by Rich Geldreich in
'1992 from an idea in Donald Knuth's TAOCP.
'
'In a normal sieve each prime found is used in turn to mark all its
'composites thus the complete sieve must be hold in memory tor the final
'tally. In Rich's version all primes found so far are used at the same
'time to mark composites in the same moving slice of ths sieve, the
'numbers left unmarked are primes,and they can be counted as the slice
'progresses.
'In fact there is no data representing the sieve slice...only a priority
'queue that keeps the primes and it's factors used in the present sieve
'slice. This queue has to be dimensioned to hold all primes up to the
'square root of the maximum prime, the present size of 4096 would allow
'for primes up to 2^31.
'Additional optimizations;
' Multiples of 2 and 3 are skipped
' A prime p starts to sieve at p*p, because p*a for a<p will be found
' by a.
' The heap is an udt but is kept in separate arrays for speed.
DEFINT AZ
DECLARE SUB PutPrime (a&)
DECLARE FUNCTION GetPrime& ()
CONST heapsize = 4096
'Priority queue
DIM heapq(1 TO heapsize) AS LONG
DIM HeapQ1(1 TO heapsize) AS LONG
DIM HeapQ2(1 TO heapsize) AS LONG
DIM SHARED n AS LONG
DIM t AS LONG
DIM Q AS LONG, Q1 AS LONG, Q2 AS LONG
DIM TQ AS LONG, TQ1 AS LONG
DIM u AS LONG, primepos AS LONG, cnt AS LONG
primepos = 1000
n = 5
d = 2
r = 1
t = 25
heapq(1) = 25
HeapQ1(1) = 10
HeapQ2(1) = 30
cnt = 2
DO
DO
Q = heapq(1)
Q1 = HeapQ1(1)
Q2 = HeapQ2(1)
TQ = Q + Q1
TQ1 = Q2  Q1
'***Insert Heap(1) into priority queue
i = 1
DO
j = i * 2
IF j <= r THEN
IF j < r THEN
IF heapq(j) > heapq(j + 1) THEN
j = j + 1
END IF
END IF
IF TQ > heapq(j) THEN
heapq(i) = heapq(j)
HeapQ1(i) = HeapQ1(j)
HeapQ2(i) = HeapQ2(j)
i = j
ELSE
EXIT DO
END IF
ELSE
EXIT DO
END IF
LOOP
heapq(i) = TQ
HeapQ1(i) = TQ1
HeapQ2(i) = Q2
'***
LOOP UNTIL n <= Q
DO WHILE n < Q
cnt = cnt + 1
IF cnt < heapsize THEN heapq(cnt  2) = n
IF cnt = primepos THEN
PRINT USING "The ####### th prime is ######### "; primepos; n
IF primepos = 100000 THEN PRINT "Ended": SYSTEM
primepos = primepos * 10
END IF
n = n + d
d = 6  d
LOOP
IF n = t THEN
u = heapq(r + 1)
t = u * u
'***Find location for new entry
j = r + 1
DO
i = j \ 2
IF i = 0 THEN
EXIT DO
END IF
IF heapq(i) <= t THEN
EXIT DO
END IF
heapq(j) = heapq(i)
HeapQ1(j) = HeapQ1(i)
HeapQ2(j) = HeapQ2(i)
j = i
LOOP
'***
heapq(j) = t
IF (u MOD 3) = 2 THEN
HeapQ1(j) = 2 * u
ELSE
HeapQ1(j) = 4 * u
END IF
HeapQ2(j) = 6 * u
r = r + 1
END IF
n = n + d
d = 6  d
LOOP
Antoni
Posts: 484
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I've just been looking at some of the previous posts about primes, seeing what methods other people used. I found a lot by you Antoni!, and some other interesting things I may try and add to my program.
I came across this too which made me laugh
http://members.surfeu.fi/kklaine/primebear.html
EVEN MEN OF STEEL RUST.
Posts: 484
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Joined: Apr 2005
Thats a nice one Antoni, works fast, i'm still trying to understand how it works.
I improved my second one, its a bit faster now, but I still need to learn more to make it go even faster. Some of the code wasn't necessary, and i even forgot to ignore multiples of 2.
Code: Const MaxPrime = 1000000 ' The max value for P
Const MaxVal = MaxPrime * 16 ' The max value for N (a bit of a cheat using * 16...)
Const Max32 = (MaxVal \ 32) + 1 ' The number of 32bit vars needed to store bit array
Dim Shared BitsArray(0 To Max32) As uInteger
Dim As uInteger n1, n2, count, p, o, n1x2, n1x3
Dim As uInteger steps(0 To 47) = { _
2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, _
6, 4, 6, 8, 4, 2, 4, 2, 4, 8, 6, 4, 6, 2, 4, 6, _
2, 6, 6, 4, 2, 4, 6, 2, 6, 4, 2, 4, 2, 10, 2, 10 _
} ' This lookup table is used to calculate the step, to avoid multiples of 2, 3, 5, and 7
' any more that that and the table becomes very large (the one inc. 11 is 480 entrys)
Dim As Integer curr_step
Dim As Integer prime_to_find = 1000
count = 4 ' start count offset to account for 2, 3, 5 and 7 being prime
n1 = 11 ' start at 11 due to count starting at 4
While n1 <= MaxVal
p = n1 shr 5 ' \ 32 'p is integer postition in bitarray
o = n1 and 31 ' mod 32 'o is bit offset
If NOT BIT(BitsArray(p), o) Then ' If the bit isn't set then it hasn't been struck out
count += 1
If count = prime_to_find Then
Print Using "###,###,###th prime  ###,###,###"; prime_to_find; n1
If prime_to_find = 1000000 Then Exit While
prime_to_find *= 10
End If
If n1 <= (sqr(MaxVal) + 1) Then ' Only strike out multiples of primes <= sqr(MaxVal)
n1x2 = n1 + n1 ' (the +1 is just to account for any rounding, may not
n1x3 = n1x2 + n1 ' be needed?)
' we don't need to step by n1, as that will wastefully look at even numbers (odd+odd=even)
' same goes for start pos, n1 is odd, so 2*n1 not needed, start at 3*n1
For n2 = n1x3 To MaxVal Step n1x2 ' work from 3n to max marking off multiples
p = n2 shr 5 ' \ 32 'p is integer postition in bitarray
o = n2 and 31 ' mod 32 'o is bit offset
BitsArray(p) = BITSET(BitsArray(p), o) ' Set the bit to show its bad
Next n2
End If
End If
' we can step by set amounts, to avoid multiples of 2, 3 and 5
n1 += steps(curr_step)
curr_step += 1
If curr_step = 48 Then curr_step = 0
Wend
EDIT: added a check for n1 <= sqr(maxval)
EVEN MEN OF STEEL RUST.
Posts: 1,407
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Quote:I came across this too which made me laugh
http://members.surfeu.fi/kklaine/primebear.html
Ok,Arktinen Krokotiili Projekti is the winner! Let's close the contest, nothing more more can be done.. :rotfl: :rotfl: :rotfl:
EDITED:
Their javascript prime finding algorithm is a little slow..
Code: function is_x_prime_number(x)
{
var limit=0;
var div=3;
var x_limit = Math.sqrt(x);
while (x%div!=0 && div<x_limit)div+=2;
is_prime = (x%div==0 && x!=div)*1
return is_prime;
}
A simple trial division...
Antoni
