\datethis @*Intro. This program constructs segments of the sieve of Erasthosthenes,'' and outputs the largest prime gaps that it finds. More precisely, it works with sets of prime numbers between $s_i$ and $s_{i+1}=s_i+\delta$, represented as an array of bits, and it examines these arrays for $t$ consecutive intervals beginning with $s_i$ for $i=0$, 1, \dots~$t-1$. Thus it scans all primes between $s_0$ and $s_t$. Let $p_k$ be the $k$th prime number. The sieve of Eratotosthenes determines all primes $\le N$ by starting with the set $\{2,3,\ldots,N\}$ and striking out the nonprimes: After we know $p_1$ through $p_{k-1}$, the next remaining element is $p_k$, and we strike out the numbers $p_k^2$, $p_k(p_k+1)$, $p_k(p_k+2)$, etc. The sieve is complete when we've found the first prime with $p_k^2>N$. In this program it's convenient to deal with the nonprimes instead of the primes, and to assume that we already know all of the small'' primes~$p_k$ for which $p_k^2\le s_t$. And of course we might as well restrict consideration to odd numbers. Thus, we'll represent the integers between $s_i$ and $s_{i+1}$ by $\delta/2$ bits; these bits will appear in $\delta/128$ 64-bit numbers |sieve[j]|, where $$|sieve[j]|=\sum_{n=s_i+128j}^{s_i+128(j+1)} 2^{(n-s_i-128j-1)/2} \,\hbox{\bigl[n is an odd multiple of some odd prime \le\sqrt{\mathstrut s_{i+1}}\,\bigr]}.$$ We choose the segment size $\delta$ to be a multiple of 128. We also assume that $s_0$ is even, and $s_0\ge\sqrt\delta$. It follows that $s_i$ is even for all~$i$, and that $(s_i+1)^2=s_i^2+s_i+s_{i+1}-\delta \ge s_i+s_{i+1}>s_{i+1}$. Consequently we have $$|sieve[j]|=\sum_{n=s_i+128j}^{s_i+128(j+1)} 2^{(n-s_i-128j-1)/2} \,\hbox{\bigl[n is odd and not prime\bigr]},$$ because $n$ appears if and only if it is divisible by some prime~$p$ where $p\le\sqrt{\mathstrut s_{i+1}}s_t$; it may also contain further primes, which are ignored. It is a binary file, with each prime given as an |unsigned int|. (There are 203,280,221 primes less than $2^{32}$, the largest of which is $2^{32}-5$. Thus I'm implicitly assuming that $s_t<(2^{32}-5)^2 \approx 1.8\times10^{19}$.) The output file is a short text file that reports large gaps. Whenever the program discovers consecutive primes for which the gap $p_{k+1}-p_k$ is greater than or equal to all previously seen gaps, this gap is output (unless it is smaller than 256). The smallest and largest primes between $s_0$ and $s_t$ are also output, so that we can keep track of gaps between primes that are found by different instances of this program. @d del 1000000000LL /* the segment size $\delta$, a multiple of 128 */ @d kmax 51000000 /* an index such that $p_{kmax}^2>s_t$ */ @c #include #include #include FILE *infile, *outfile; unsigned int prime[kmax]; /* $|prime|[k]=p_{k+1}$ */ unsigned int start[kmax]; /* indices for initializing a segment */ unsigned long long sieve[2+del/128]; unsigned long long s0; /* beginning of the first segment */ int tt; /* number of segments */ unsigned long long st; /* ending of the last segment */ unsigned long long lastprime; /* largest prime so far, if any */ int bestgap=256; /* lower bound for gap reporting */ unsigned long long sv[11]; /* bit patterns for the smallest primes */ int rem[11]; /* shift amounts for the smallest primes */ char nu[0x10000]; /* table for counting bits */ int timer; main(int argc, char*argv[]) { register j,k; unsigned long long x,y,z,s,ss; int d,ii,kk; timer=time(0); @; @; @; for (ii=0;ii; @; printf("(Finished; the last segment took %d sec.)\n",time(0)-timer); } @ @= if (argc!=5 || sscanf(argv[1],"%llu",&s0)!=1 || sscanf(argv[2],"%d",&tt)!=1) { fprintf(stderr,"Usage: %s s[0] t inputfile outputfile\n",argv[0]); exit(-1); } infile=fopen(argv[3],"rb"); if (!infile) { fprintf(stderr,"I can't open %s for binary input!\n",argv[3]); exit(-2); } outfile=fopen(argv[4],"w"); if (!outfile) { fprintf(stderr,"I can't open %s for text output!\n",argv[4]); exit(-3); } st=s0+tt*del; if (del%128) { fprintf(stderr,"Oops: The sieve size %d isn't a multiple of 128!\n",del); exit(-4); } if (s0&1) { fprintf(stderr, "The starting point %llu isn't even!\n", s0); exit(-5); } if (s0*s0; printf("Sieving between s[0]=%llu and s[t]=%llu:\n",s0,st); @ @= for (k=0;;k++) { if (k>=kmax) { fprintf(stderr,"Oops: Please recompile me with kmax>%d!\n",kmax); exit(-7); } if (fread(&prime[k],sizeof(unsigned int),1,infile)!=1) { fprintf(stderr,"The input file ended prematurely (%d^2<%llu)!\n", k? prime[k-1]: 0,st); exit(-8); } if (k==0 && prime[0]!=2) { fprintf(stderr,"The input file begins with %d, not 2!\n",prime[0]); exit(-9); } else if (k>0 && prime[k]<=prime[k-1]) { fprintf(stderr,"The input file has consecutive entries %d,%d!\n", prime[k-1],prime[k]); exit(-10); } if (((unsigned long long)prime[k])*prime[k]>st) break; } printf("%d primes successfully loaded from %s\n",k,argv[3]); @*Sieving. Let's say that the prime $p_k$ is active'' if $p_k^2= for (k=1;((unsigned long long)prime[k])*prime[k]>1); else start[k]=(prime[k]-j-1)>>1; } kk=k; @; @ Primes less than 32 will appear at least twice in every octabyte of the sieve. So we handle them in a slightly more efficient way, unless they're initially inactive. @= for (k=1;prime[k]<32 && k= @; ss=s0; /* base address of the next segment */ sieve[1+del/128]=-1; /* store a sentinel */ @ @= { s=ss, ss=s+del; /*$s=s_i$,$|ss|=s_{i+1}$*/ j=time(0); printf("Beginning segment %llu (after %d sec)\n",s,j-timer); fflush(stdout); timer=j; @; @; @; @; } @ @= for (j=0;j>rem[k]); } sieve[j]=z; } @ Now we want to set 1 bits for every odd multiple of |prime[k]| in the current segment, whenever |prime[k]| is active. The bit for the integer$s_i+2j+1$is |1<<(j&0x3f)| in |sieve[j>>6]|, for$0\le j<\delta/2$. @= for (k=d;k>6] |= 1LL<<(j&0x3f); start[k]=j-del/2; } @ @= while (((unsigned long long)prime[k])*prime[k]>1;j>6] |= 1LL<<(j&0x3f); start[k]=j-del/2; k++; } kk=k; @* Processing gaps. If$p_{k+1}-p_k\ge256$, we're bound to find an octabyte of all 1s in the sieve between the 0~for~$p_k$and the 0~for~$p_{k+1}$. In such cases, we check to see if this gap breaks or ties the current record. Complications occur if the gap appears at the very beginning or end of a segment, or if an entire segment is prime-free. I've tried to get the logic correct, without slowing the program down. But if any bugs are present in this code, I suppose they are due to a fallacy in this aspect of my reasoning. Two sentinels appear at the end of the sieve, in order to speed up loop termination: |sieve[del/128]=0| and |sieve[1+del/128]=-1|. @= j=0; @; while (1) { /* at this point |j; }@+else@+{ /* |j=1+del/128| and |sieve[del/128-1]!=-1| */ k=del/128-1;@+break; } } @; donewithseg:@; @ We don't need to figure out the exact value of the first prime greater than~|s| unless the present segment begins with an octabyte of all 1s, or the previous segment ends with such an octabyte, or we're in the first segment. But in any case we'll want to go immediately to |donewithseg| if the current segment is entirely prime-free. And we always want to end this step with |j| equal to the smallest index such that |sieve[j]!=-1|. @= if (lastprime<=s-128 || sieve[j]==-1) { for (;sieve[j]==-1;j++); if (j==del/128) goto donewithseg; y=~sieve[j]; y=y&-y; /* extract the rightmost 1 bit */ @; x=s+(j<<7)+y+y+1; /* this is the first prime of the segment */ if (lastprime) @@; else { k=x-s0; fprintf(outfile,"The first prime is %llu = s[0]+%d\n",x,k); fflush(outfile); } } @ When |sieve[k]!=-1| and |sieve[j]!=-1| and everything between them is |-1| (all ones), there's a gap of size~$g$where$128\vert j-k\vert-126\le g\le128\vert j-k\vert+126\$. @= if (((j-k)<<7)+126>=bestgap) { y=~sieve[j]; y=y&-y; /* extract the rightmost 1 bit */ @; x=s+(j<<7)+y+y+1; /* this is the first prime after the gap */ @; @; } @ @= for (y=~sieve[k],z=y&(y-1);z;y=z,z=y&(y-1)); @; lastprime=s+(k<<7)+y+y+1; @ As far as I know, the following method is the fastest way to compute binary logarithms on an Opteron computer (which is the machine I'm targeting here). @= y--; y=nu[y&0xffff]+nu[(y>>16)&0xffff]+nu[(y>>32)&0xffff]+nu[(y>>48)&0xffff]; @ With a more extensive table, I could count the 1s in an arbitrary binary word. But seventeen table entries are sufficient for present purposes. @= for (j=0;j<=16;j++) nu[((1<= { if (x-lastprime>=bestgap) { bestgap=x-lastprime; fprintf(outfile,"%llu is followed by a gap of length %d\n", lastprime,bestgap); fflush(outfile); } } @ @= if (lastprime) { k=st-lastprime; fprintf(outfile,"The final prime is %llu = s[t]-%d.\n",lastprime,k); }@+else fprintf(outfile,"No prime numbers exist between s[0] and s[t].\n"); @*Index.