diff --git a/basis/math/primes/miller-rabin/miller-rabin-docs.factor b/basis/math/primes/miller-rabin/miller-rabin-docs.factor
index 2455dafdd5..2d19d51e06 100644
--- a/basis/math/primes/miller-rabin/miller-rabin-docs.factor
+++ b/basis/math/primes/miller-rabin/miller-rabin-docs.factor
@@ -3,20 +3,6 @@
 USING: help.markup help.syntax kernel sequences math ;
 IN: math.primes.miller-rabin
 
-HELP: find-relative-prime
-{ $values
-    { "n" integer }
-    { "p" integer }
-}
-{ $description "Returns a number that is relatively prime to " { $snippet "n" } "." } ;
-
-HELP: find-relative-prime*
-{ $values
-    { "n" integer } { "guess" integer }
-    { "p" integer }
-}
-{ $description "Returns a number that is relatively prime to " { $snippet "n" } ", starting by trying " { $snippet "guess" } "." } ;
-
 HELP: miller-rabin
 { $values
     { "n" integer }
@@ -33,68 +19,10 @@ HELP: miller-rabin*
 }
 { $description "Performs " { $snippet "numtrials" } " trials of the Miller-Rabin probabilistic primality test algorithm and returns true if prime." } ;
 
-HELP: next-prime
-{ $values
-    { "n" integer }
-    { "p" integer }
-}
-{ $description "Tests consecutive numbers for primality with " { $link miller-rabin } " and returns the next prime." } ;
-
-HELP: next-safe-prime
-{ $values
-    { "n" integer }
-    { "q" integer }
-}
-{ $description "Tests consecutive numbers and returns the next safe prime. A safe prime is desirable in cryptography applications such as Diffie-Hellman and SRP6." } ;
-
-HELP: random-bits*
-{ $values
-    { "numbits" integer }
-    { "n" integer }
-}
-{ $description "Returns an integer exactly " { $snippet "numbits" } " in length, with the topmost bit set to one." } ;
-
-HELP: random-prime
-{ $values
-    { "numbits" integer }
-    { "p" integer }
-}
-{ $description "Returns a prime number exactly " { $snippet "numbits" } " bits in length, with the topmost bit set to one." } ;
-
-HELP: random-safe-prime
-{ $values
-    { "numbits" integer }
-    { "p" integer }
-}
-{ $description "Returns a safe prime number " { $snippet "numbits" } " bits in length, with the topmost bit set to one." } ;
-
-HELP: safe-prime?
-{ $values
-    { "q" integer }
-    { "?" "a boolean" }
-}
-{ $description "Tests whether the number is a safe prime. A safe prime " { $snippet "p" } " must be prime, as must " { $snippet "(p - 1) / 2" } "." } ;
-
-HELP: unique-primes
-{ $values
-    { "numbits" integer } { "n" integer }
-    { "seq" sequence }
-}
-{ $description "Generates a sequence of " { $snippet "n" } " unique prime numbers with exactly " { $snippet "numbits" } " bits." } ;
-
 ARTICLE: "math.primes.miller-rabin" "Miller-Rabin probabilistic primality test"
 "The " { $vocab-link "math.primes.miller-rabin" } " vocabulary implements the Miller-Rabin probabilistic primality test and utility words that use it in order to generate random prime numbers." $nl
 "The Miller-Rabin probabilistic primality test:"
 { $subsection miller-rabin }
-{ $subsection miller-rabin* }
-"Generating relative prime numbers:"
-{ $subsection find-relative-prime }
-{ $subsection find-relative-prime* }
-"Generating prime numbers:"
-{ $subsection next-prime }
-{ $subsection random-prime }
-"Generating safe prime numbers:"
-{ $subsection next-safe-prime }
-{ $subsection random-safe-prime } ;
+{ $subsection miller-rabin* } ;
 
 ABOUT: "math.primes.miller-rabin"
diff --git a/basis/math/primes/miller-rabin/miller-rabin-tests.factor b/basis/math/primes/miller-rabin/miller-rabin-tests.factor
index 9c635c8f38..aeae6cac1b 100644
--- a/basis/math/primes/miller-rabin/miller-rabin-tests.factor
+++ b/basis/math/primes/miller-rabin/miller-rabin-tests.factor
@@ -1,5 +1,6 @@
-USING: math.primes.miller-rabin tools.test kernel sequences
-math.primes.miller-rabin.private math ;
+USING: kernel math math.primes math.primes.miller-rabin
+math.primes.miller-rabin.private math.primes.safe
+math.primes.safe.private random sequences tools.test ;
 IN: math.primes.miller-rabin.tests
 
 [ f ] [ 473155932665450549999756893736999469773678960651272093993257221235459777950185377130233556540099119926369437865330559863 miller-rabin ] unit-test
diff --git a/basis/math/primes/miller-rabin/miller-rabin.factor b/basis/math/primes/miller-rabin/miller-rabin.factor
index 35ee97a897..b0dfc4ed35 100755
--- a/basis/math/primes/miller-rabin/miller-rabin.factor
+++ b/basis/math/primes/miller-rabin/miller-rabin.factor
@@ -1,18 +1,9 @@
 ! Copyright (c) 2008-2009 Doug Coleman.
 ! See http://factorcode.org/license.txt for BSD license.
-USING: combinators kernel locals math math.functions math.ranges
-random sequences sets combinators.short-circuit math.bitwise
-math math.order ;
+USING: combinators combinators.short-circuit kernel locals math
+math.functions math.ranges random sequences sets ;
 IN: math.primes.miller-rabin
 
-: >odd ( n -- int ) 0 set-bit ; foldable
-
-: >even ( n -- int ) 0 clear-bit ; foldable
-
-: next-even ( m -- n ) >even 2 + ;
-
-: next-odd ( m -- n ) dup even? [ 1 + ] [ 2 + ] if ;
-
 <PRIVATE
 
 :: (miller-rabin) ( n trials -- ? )
@@ -42,73 +33,3 @@ PRIVATE>
     } cond ;
 
 : miller-rabin ( n -- ? ) 10 miller-rabin* ;
-
-ERROR: prime-range-error n ;
-
-: next-prime ( n -- p )
-    dup 1 < [ prime-range-error ] when
-    dup 1 = [
-        drop 2
-    ] [
-        next-odd dup miller-rabin [ next-prime ] unless
-    ] if ;
-
-: random-bits* ( numbits -- n )
-    1 - [ random-bits ] keep set-bit ;
-
-: random-prime ( numbits -- p )
-    random-bits* next-prime ;
-
-ERROR: no-relative-prime n ;
-
-<PRIVATE
-
-: (find-relative-prime) ( n guess -- p )
-    over 1 <= [ over no-relative-prime ] when
-    dup 1 <= [ drop 3 ] when
-    2dup gcd nip 1 > [ 2 + (find-relative-prime) ] [ nip ] if ;
-
-PRIVATE>
-
-: find-relative-prime* ( n guess -- p )
-    #! find a prime relative to n with initial guess
-    >odd (find-relative-prime) ;
-
-: find-relative-prime ( n -- p )
-    dup random find-relative-prime* ;
-
-ERROR: too-few-primes ;
-
-: unique-primes ( numbits n -- seq )
-    #! generate two primes
-    swap
-    dup 5 < [ too-few-primes ] when
-    2dup [ random-prime ] curry replicate
-    dup all-unique? [ 2nip ] [ drop unique-primes ] if ;
-
-! Safe primes are of the form p = 2q + 1, p,q are prime
-! See http://en.wikipedia.org/wiki/Safe_prime
-
-<PRIVATE
-
-: safe-prime-candidate? ( n -- ? )
-    1 + 6 divisor? ;
-
-: next-safe-prime-candidate ( n -- candidate )
-    next-prime dup safe-prime-candidate?
-    [ next-safe-prime-candidate ] unless ;
-
-PRIVATE>
-
-: safe-prime? ( q -- ? )
-    {
-        [ 1 - 2 / dup integer? [ miller-rabin ] [ drop f ] if ]
-        [ miller-rabin ]
-    } 1&& ;
-
-: next-safe-prime ( n -- q )
-    next-safe-prime-candidate
-    dup safe-prime? [ next-safe-prime ] unless ;
-
-: random-safe-prime ( numbits -- p )
-    random-bits* next-safe-prime ;
diff --git a/basis/math/primes/primes-docs.factor b/basis/math/primes/primes-docs.factor
index c7dbc950e8..fa991e800f 100644
--- a/basis/math/primes/primes-docs.factor
+++ b/basis/math/primes/primes-docs.factor
@@ -1,10 +1,10 @@
-USING: help.markup help.syntax ;
+USING: help.markup help.syntax math sequences ;
 IN: math.primes
 
 { next-prime prime? } related-words
 
 HELP: next-prime
-{ $values { "n" "an integer not smaller than 2" } { "p" "a prime number" } }
+{ $values { "n" integer } { "p" "a prime number" } }
 { $description "Return the next prime number greater than " { $snippet "n" } "." } ;
 
 HELP: prime?
@@ -20,3 +20,49 @@ HELP: primes-upto
 HELP: primes-between
 { $values { "low" "an integer" } { "high" "an integer" } { "seq" "a sequence" } }
 { $description "Return a sequence containing all the prime numbers between " { $snippet "low" } " and " { $snippet "high" } "." } ;
+
+HELP: find-relative-prime
+{ $values
+    { "n" integer }
+    { "p" integer }
+}
+{ $description "Returns a number that is relatively prime to " { $snippet "n" } "." } ;
+
+HELP: find-relative-prime*
+{ $values
+    { "n" integer } { "guess" integer }
+    { "p" integer }
+}
+{ $description "Returns a number that is relatively prime to " { $snippet "n" } ", starting by trying " { $snippet "guess" } "." } ;
+
+HELP: random-prime
+{ $values
+    { "numbits" integer }
+    { "p" integer }
+}
+{ $description "Returns a prime number exactly " { $snippet "numbits" } " bits in length, with the topmost bit set to one." } ;
+
+HELP: unique-primes
+{ $values
+    { "numbits" integer } { "n" integer }
+    { "seq" sequence }
+}
+{ $description "Generates a sequence of " { $snippet "n" } " unique prime numbers with exactly " { $snippet "numbits" } " bits." } ;
+
+
+ARTICLE: "math.primes" "Prime numbers"
+"The " { $vocab-link "math.primes" } " vocabulary implements words related to prime numbers." $nl
+"Testing if a number is prime:"
+{ $subsection prime? }
+"Generating prime numbers:"
+{ $subsection next-prime }
+{ $subsection primes-upto }
+{ $subsection primes-between }
+{ $subsection random-prime }
+"Generating relative prime numbers:"
+{ $subsection find-relative-prime }
+{ $subsection find-relative-prime* }
+"Make a sequence of random prime numbers:"
+{ $subsection unique-primes } ;
+
+ABOUT: "math.primes"
diff --git a/basis/math/primes/primes-tests.factor b/basis/math/primes/primes-tests.factor
index db738399ef..6580f0780e 100644
--- a/basis/math/primes/primes-tests.factor
+++ b/basis/math/primes/primes-tests.factor
@@ -1,4 +1,6 @@
-USING: arrays math.primes tools.test ;
+USING: arrays math math.primes math.primes.miller-rabin
+tools.test ;
+IN: math.primes.tests
 
 { 1237 } [ 1234 next-prime ] unit-test
 { f t } [ 1234 prime? 1237 prime? ] unit-test
@@ -7,3 +9,12 @@ USING: arrays math.primes tools.test ;
 
 { { 4999963 4999999 5000011 5000077 5000081 } }
 [ 4999962 5000082 primes-between >array ] unit-test
+
+[ 2 ] [ 1 next-prime ] unit-test
+[ 3 ] [ 2 next-prime ] unit-test
+[ 5 ] [ 3 next-prime ] unit-test
+[ 101 ] [ 100 next-prime ] unit-test
+[ t ] [ 2135623355842621559 miller-rabin ] unit-test
+[ 100000000000031 ] [ 100000000000000 next-prime ] unit-test
+
+[ 49 ] [ 50 random-prime log2 ] unit-test
diff --git a/basis/math/primes/primes.factor b/basis/math/primes/primes.factor
index fa1cd5cb63..e3985fc600 100644
--- a/basis/math/primes/primes.factor
+++ b/basis/math/primes/primes.factor
@@ -1,8 +1,8 @@
 ! Copyright (C) 2007-2009 Samuel Tardieu.
 ! See http://factorcode.org/license.txt for BSD license.
-USING: combinators kernel math math.functions
-math.primes.miller-rabin math.order math.primes.erato
-math.ranges sequences ;
+USING: combinators kernel math math.bitwise math.functions
+math.order math.primes.erato math.primes.miller-rabin
+math.ranges random sequences sets fry ;
 IN: math.primes
 
 <PRIVATE
@@ -22,7 +22,11 @@ PRIVATE>
     } cond ; foldable
 
 : next-prime ( n -- p )
-    next-odd [ dup really-prime? ] [ 2 + ] until ; foldable
+    dup 2 < [
+        drop 2
+    ] [
+        next-odd [ dup really-prime? ] [ 2 + ] until
+    ] if ; foldable
 
 : primes-between ( low high -- seq )
     [ dup 3 max dup even? [ 1 + ] when ] dip
@@ -32,3 +36,34 @@ PRIVATE>
 : primes-upto ( n -- seq ) 2 swap primes-between ;
 
 : coprime? ( a b -- ? ) gcd nip 1 = ; foldable
+
+: random-prime ( numbits -- p )
+    random-bits* next-prime ;
+
+: estimated-primes ( m -- n )
+    dup log / ; foldable
+
+ERROR: no-relative-prime n ;
+
+<PRIVATE
+
+: (find-relative-prime) ( n guess -- p )
+    over 1 <= [ over no-relative-prime ] when
+    dup 1 <= [ drop 3 ] when
+    2dup gcd nip 1 > [ 2 + (find-relative-prime) ] [ nip ] if ;
+
+PRIVATE>
+
+: find-relative-prime* ( n guess -- p )
+    #! find a prime relative to n with initial guess
+    >odd (find-relative-prime) ;
+
+: find-relative-prime ( n -- p )
+    dup random find-relative-prime* ;
+
+ERROR: too-few-primes n numbits ;
+
+: unique-primes ( n numbits -- seq )
+    2dup 2^ estimated-primes > [ too-few-primes ] when
+    2dup '[ _ random-prime ] replicate
+    dup all-unique? [ 2nip ] [ drop unique-primes ] if ;