We have looked at the “sieve” before, and realistically it is actually a good candidate for recursion.

Below is the main (calling) program in Fortran. The program uses a sieve which is stored as a dynamic array, by declaring the array `primes`

as `allocatable`

. This makes it more efficient from a storage perspective. The user can input a value for the upper bound to check for primes, and `allocate()`

is called to create the associated resources for `prime`

. After the recursive subroutine `sieve()`

is called, the remainder of the program deals with printing the primes to a file, `primes.txt`

, in an orderly, column-based manner.

```
program eratosthenes
integer :: i, j, N
integer, dimension(:), allocatable :: primes
open(unit=9,file='primes.txt',status='replace',action='write')
write (*,*) "Enter the boundary to check for primes: "
read (*,*) N
allocate(primes(N))
do i = 1, N
primes(i) = i
end do
call sieve(N,primes,2)
write (*,50) "Prime numbers from 1 to ", N, " written to file primes.txt"
50 format (A,I8,A)
j = 1
do i = 1, N
if (primes(i) /= 0) then
write (9,"(I5)",advance="no") primes(i)
j = j + 1
if (mod(j,10) == 0) then
write (9,*)
end if
end if
end do
close (9,status='keep')
end program eratosthenes
```

Now, on to the recursive subroutine `sieve()`

. The subroutine has three parameters: `N`

, the size of the sieve, `p`

, the sieve array holding the primes, and `x`

, the starting value, in this case 2.

```
recursive subroutine sieve(N, p, x)
integer, intent(in) :: N
integer, intent(inout), dimension(N) :: p
integer, intent(in) :: x
integer :: j, k
k = sqrt(real(N))
if (x == 2) then
do j = 4, N, 2
p(j) = 0
end do
call sieve(N,p,x+1)
elseif (mod(x,2) == 1 .and. x <= k) then
do j = x, N, x
!if (mod(p(j),x) == 0 .and. p(j) /= x) then
if (p(j) /= x) then
p(j) = 0
end if
end do
call sieve(N,p,x+2)
end if
end subroutine sieve
```

What is happening in this subroutine?

- On Line 10, if the value of
`x`

is 2, then all even numbers except 2 are set to 0. After this,`sieve()`

is called with the`x+1`

. This section of code is only actioned in the initial call to`sieve()`

. - On Line 15, the odd values are processed, until
`x`

is <=`k`

. All multiples of`j`

are marked in`p`

by setting them to zero. For example if`x`

= 3, then 6, 9, 12, etc. are set to 0. At the end of this section of code (Line 22),`sieve()`

is called with`x+2`

, making sure only odd numbers are processed.

Here is the output from the program:

1 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 211 223 227 229 233 239 241 251 257 263 269 271 277 281 283 293 307 311 313 317 331 337 347 349 353 359 367 373 379 383 389 397 401 409 419 421 431 433 439 443 449 457 461 463 467 479 487 491 499 503 509 521 523 541 547 557 563 569 571 577 587 593 599 601 607 613 617 619 631 641 643 647 653 659 661 673 677 683 691 701 709 719 727 733 739 743 751 757 761 769 773 787 797 809 811 821 823 827 829 839 853 857 859 863 877 881 883 887 907 911 919 929 937 941 947 953 967 971 977 983 991 997