Published at Oct 30 2019
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Instructions

Test suite

Solution

Use the Sieve of Eratosthenes to find all the primes from 2 up to a given number.

The Sieve of Eratosthenes is a simple, ancient algorithm for finding all prime numbers up to any given limit. It does so by iteratively marking as composite (i.e. not prime) the multiples of each prime, starting with the multiples of 2. It does not use any division or remainder operation.

Create your range, starting at two and continuing up to and including the given limit. (i.e. [2, limit])

The algorithm consists of repeating the following over and over:

- take the next available unmarked number in your list (it is prime)
- mark all the multiples of that number (they are not prime)

Repeat until you have processed each number in your range.

When the algorithm terminates, all the numbers in the list that have not been marked are prime.

The wikipedia article has a useful graphic that explains the algorithm: https://en.wikipedia.org/wiki/Sieve_of_Eratosthenes

Notice that this is a very specific algorithm, and the tests don't check that you've implemented the algorithm, only that you've come up with the correct list of primes. A good first test is to check that you do not use division or remainder operations (div, /, mod or % depending on the language).

Check out Installing Common Lisp for instructions to get started or take a look at the guides available in the track's side bar.

While Common Lisp doesn't care about indentation and layout of code, nor whether you use spaces or tabs, this is an important consideration for submissions to exercism.io. Excercism.io's code widget cannot handle mixing of tab and space characters well so using only spaces is recommended to make the code more readable to the human reviewers. Please review your editors settings on how to accomplish this. Below are instructions for popular editors for Common Lisp.

Use the following commands to ensure VIM uses only spaces for indentation:

```
:set tabstop=2
:set shiftwidth=2
:set expandtab
```

(or as a oneliner `:set tabstop=2 shiftwidth=2 expandtab`

). This can
be added to your `~/.vimrc`

file to use it all the time.

Emacs is very well suited for editing Common Lisp and has many powerful add-on packages available. The only thing that one needs to do with a stock emacs to make it work well with exercism.io is to evaluate the following code:

`(setq-default indent-tabs-mode nil)`

This can be placed in your `~/.emacs`

(or `~/.emacs.d/init.el`

) in
order to have it set whenever Emacs is launched.

One suggested add-on for Emacs and Common Lisp is SLIME which offers tight integration with the REPL; making iterative coding and testing very easy.

Sieve of Eratosthenes at Wikipedia http://en.wikipedia.org/wiki/Sieve_of_Eratosthenes

It's possible to submit an incomplete solution so you can see how others have completed the exercise.

```
(ql:quickload "lisp-unit")
#-xlisp-test (load "sieve")
(defpackage #:sieve-test
(:use #:common-lisp #:lisp-unit))
(in-package #:sieve-test)
(define-test no-primes-under-two
(assert-equal '()
(sieve:primes-to 1)))
(define-test find-first-prime
(assert-equal '(2)
(sieve:primes-to 2)))
(define-test find-primes-up-to-10
(assert-equal '(2 3 5 7)
(sieve:primes-to 10)))
(define-test limit-is-prime
(assert-equal '(2 3 5 7 11 13)
(sieve:primes-to 13)))
(define-test primes-below-1000
(let ((primes-below-1000
'( 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)))
(assert-equal primes-below-1000
(sieve:primes-to 1000))))
#-xlisp-test
(let ((*print-errors* t)
(*print-failures* t))
(run-tests :all :sieve-test))
```

```
(defpackage #:sieve
(:use #:cl)
(:export #:primes-to)
(:documentation "Generates a list of primes up to a given limit."))
(in-package #:sieve)
(defun numbers-to (n)
(labels
((f (n acc)
(if (= n 1)
acc
(f (1- n) (cons n acc)))))
(f n '())))
(defun primep (p)
(labels
((f (p trials chance)
(let ((a (1+ (random (1- p)))))
(cond
((= trials 0) (> chance 0))
((> (gcd a p) 1) nil)
((> (rem (expt a (1- p)) p) 1) nil)
(t (f p (1- trials) (1+ chance)))))))
(f p 20 0)))
(defun primes-to (n)
(remove-if-not #'primep (numbers-to n)))
```

A huge amount can be learned from reading other people’s code. This is why we wanted to give exercism users the option of making their solutions public.

Here are some questions to help you reflect on this solution and learn the most from it.

- What compromises have been made?
- Are there new concepts here that you could read more about to improve your understanding?

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