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Pseudoprimes Roland Engdahl University of Kalmar Sweden mr.engdahl@telia.com

In order to prove primality of a number, previously we required a list of primes. Eratosthenes, about 230 B.C., created his famous sieve, The Sieve of Eratosthenes. Nowadays we have methods to perform sieve and store a table of primes on a computer system. Fermat, 1640 , described one characteristic of primes in his famous theorem. The converse of this theorem is however not true. There are composite numbers which comply with this characteristic These numbers are called pseudoprimes. This application provides examples and programs about primes, pseudoprimes, Carmichael numbers, strong pseudoprimes and primality testing. Only definitions of the concepts are included. Proofs and deeper studies in the theory are to be found in books on number theory e g Kumanduri and Romero, Number theory with computer applications. A new paradigm in mathematics is experimenting. Consult these sources: Borwein and Bailey and Girgensohn, (vol 1) Mathematics by Experiment, Plausible Reasoning in the 21 st Century, 2004. (vol 2) Experimentation in Mathematics, Computational Paths to Discovery, 2004. Perhaps experiments with these programs will increase the esteem of number theory.

If p is prime then 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

5 is a prime. Divide 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

97 is a prime Calculate LUkkbW9kRzYiNiQtSSJeRyUqcHJvdGVjdGVkRzYkIiU2NiIjJyoiIygq

1111&^96 mod 97;

IiIi

Calculate LUkkbW9kRzYiNiQtSSJeRyUqcHJvdGVjdGVkRzYkIiIjIiRTJCIkVCQ=

2&^340 mod 341;

IiIi

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

From the example above we see that the conversion of Fermat's theorem is false. 341 has a characteristic for a prime but is composite 341 is called a pseudoprime to base 2

A composite number n is called a pseudoprime to base a if LUkkbW9kRzYiNiQtSSgmZXF1aXY7R0YkNiQpSSJhR0YkLCZJIm5HRiQiIiIhIiJGLUYtRiw=

restart: a:=2:lim:=1000: the program calculates all pseudoprimes less than lim to base a for n from 5 by 2 to lim do if a&^(n-1) mod n = 1 and isprime(n)=false satisfies Fermat's theorem and 'isprime(n)=false' means primes excluded then print(n); end if; end do;

IiRUJA==

IiRoJg==

IiRYJw==

restart: the program calculates the number of primes and pseudoprimes between limits to a given base stt:=10000:stp:=30000:a:=2: choose start >2 stt and stop stp and base a ps:=0:pr:=0:if type(stt,even) then stt:=stt+1: end if: start with odd number for n from stt by 2 to stp do if isprime(n)=true then pr:=pr+1: end if: counts the number of primes if isprime(n)=false and a&^(n-1) mod n=1 then ps:=ps+1: counts only the pseudoprimes end if: end do: print("number of primes=",pr);print("number of pseudoprimes to base " ,a," = ",ps);

NiRRMm51bWJlcn5vZn5wcmltZXM9NiIiJTs/

NiZRQG51bWJlcn5vZn5wc2V1ZG9wcmltZXN+dG9+YmFzZX42IiIiI1Ekfj1+RiQiIz0=

The test shows that pseudoprimes are rather common. It is not a very good idea to use pseudoprimes for primality testing

restart: n:=561: testbase:=[2,3,5,7,9,11,13,15,17,19,21,23]:psp:="n psp base %a ": testing n=561 as pseudoprime in bases of testbase m:=n-1: for a in testbase do if (a&^m mod n =1) then printf(psp,a): end if: end do;

n psp base 2 n psp base 5 n psp base 7 n psp base 13 n psp base 19 n psp base 23

561 is a pseudoprime to all bases a with (561,a)=1 561=3\302\26711\302\26717. Therefore the numbers 3,9,11,15,17,21 are excluded from the bases 561 is an example of Carmichael number: See next section

A composite number n is called a Carmichael number if 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

A composite number n is a Carmichael number if and only if for every prime divisor p to n we have p-1 is divisor to n-1

restart:p[1]:=2: i:=1: e:=200: choose the highest prime factor < e for k from 3 by 2 to e do if isprime(k) then i:=i+1: p[i]:=k end if; end do: for r from 3 to i do for s from 2 to r-1 do for t from 1 to s-1 do n:=p[r]*p[s]*p[t]: m:=n-1: for i=r,s,t we have: p[i]-1 is divisor to m=n-1 if (m mod (p[r]-1) =0) and (m mod (p[s]-1) =0) and (m mod (p[t]-1) =0) then print(p[t],p[s],p[r] ,n): end if: end do : end do: end do: PLEASE WAIT UNTIL READY

NiYiIiQiIzYiIzwiJGgm

NiYiIiYiIzgiIzwiJTA2

NiYiIigiIzgiIz4iJUg8

NiYiIiYiIzwiI0giJWxD

NiYiIigiIzgiI0oiJUBH

NiYiIigiI0IiI1QiJSxt

NiYiIzgiI1AiI2giJlQkSA==

NiYiIigiIz4iI24iJTYqKQ==

NiYiIiYiI0giI3QiJiZlNQ==

NiYiIigiI0oiI3QiJlRlIg==

NiYiIzgiI1AiIygqIiZkbSU=

NiYiI1QiI2giJCwiIicsRUQ=

NiYiIigiI3QiJC4iIiZMRSY=

NiYiI1AiI3QiJDQiIic0V0g=

NiYiI1QiI3QiJFAiIidUK1Q=

NiYiI1AiI3QiJCI9IiciKSkpWw==

restart:p[1]:=2: i:=1: e:=62: for k from 3 by 2 to e do calculates the primes < e if isprime(k) then i:=i+1: p[i]:=k end if; end do: for u from 4 to i do for r from 3 to u-1 do for s from 2 to r-1 do for t from 1 to s-1 do n:=p[r]*p[s]*p[t]*p[u]: m:=n-1: for k=t,s,r,u p[k]-1 divisor to m=n-1 if (m mod (p[r]-1) =0) and (m mod (p[s]-1) =0) and (m mod (p[t]-1) =0) and (m mod (p[u]-1)=0) then print(p[t],p[s],p[r],p[u]," = ",n): end if: end do : end do: end do:end do: PLEASE WAIT UNTIL READY

NigiIzYiIzgiIzwiI0pRJX49fn42IiImaGAo

NigiIigiIzgiIz4iI1BRJX49fn42IiImdFIn

NigiIigiIzYiIzgiI1RRJX49fn42IiImVDUl

NigiIigiIzgiI0oiI2hRJX49fn42IiInIjNzIg==

NigiIzgiIzwiI1QiI2hRJX49fn42IiInQEZi

NigiIzYiI0oiI1QiI2hRJX49fn42IiInVEcmKQ==

Definition Given an odd composite integer n with n - 1 = q\302\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 n is a strong pseudorprime to base a if LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2J1EhRicvJSVib2xkR1EldHJ1ZUYnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1ElYm9sZEYnLyUrZm9udHdlaWdodEdGNy1JJW1zdXBHRiQ2JS1GLDYoUSJhRicvJSdmYW1pbHlHUSZTZXJpZkYnRi9GMkY1RjgtRiM2Iy1GLDYoUSJxRidGQEYvRjJGNUY4LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0Yr \342\211\241 1 mod n or 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 Strong pseudoprimes are introduced to get a better tool for identifying probable primes

restart: n:=12403: a:=3: try numbers n 314821 873181 76491 597871 112141 87913 12403 try bases a 2 3 5 7 11 13 spsp:=false:m:=n-1:q:=m:r:=0: writing m=n-1 as m=q*KSIiI0kickc2Ig== while type(q,even) do q:=iquo(q,2):r:=r+1: end do: e:=a&^q mod n: testing if LUkkbW9kRzYiNiQtSSgmZXF1aXY7R0YkNiQpSSJhR0YkSSJxR0YkIiIiSSJuR0Yk if e=1 then spsp:=true: end if: for i from 0 to r-1 do testing if 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 if e=m then spsp:=true: end if: e:=e*e mod n: squaring to take next i instead of exponentiation end do: print(spsp);

IiIi

SSV0cnVlRyUqcHJvdGVjdGVkRw==

if spsp then print(n,"is strong pseudoprime to base",a);end if;

NiUiJi5DIlE+aXN+c3Ryb25nfnBzZXVkb3ByaW1lfnRvfmJhc2U2IiIiJA==

restart: LL:=3: low limit odd number > 1 HL:=100000: high limit a:=11: base Digits:=20: str:=0: ps:=0: for n from LL by 2 to HL do if isprime(n)=false then spsp:=false:psp:=false: m:=n-1: q:=m: r:=0: while type(q,even) do q:=iquo(q,2):r:=r+1: end do: e:=a&^q mod n: if e=1 then str:=str+1:spsp:=true: end if; for i from 0 to r-1 do if e=m then spsp:=true:str:=str+1: end if: e:=e*e mod n: end do: if e=1 then ps:=ps+1: end if:end if: end do: print("number of strong pseudoprimes",str); print("number of pseudoprimes",ps); PLEASE WAIT

NiRRPm51bWJlcn5vZn5zdHJvbmd+cHNldWRvcHJpbWVzNiIiI0Q=

NiRRN251bWJlcn5vZn5wc2V1ZG9wcmltZXM2IiIjeg==

restart: Testing for pseudoprimes and strong pseudoprimes for number n to choose and bases in textbase to choose Digits:=20:n:=314821:testbase :=[2,3,5,7,11,13,17]: sp:="n strong pseudoprime base %a ":p:="n pseudoprime base %a ": m:=n-1:q:=m:r:=0: while type(q,even) do q:=iquo(q,2):r:=r+1: end do: for a in testbase do spsp:=false:psp:=false: e:=a&^q mod n: if e=1 then spsp:=true: printf(sp,a); end if; for i from 0 to r-1 do if e=m then spsp:=true:printf(sp,a); end if; e:=e*e mod n: end do: if e=1 and spsp=false then psp:=true: printf(p,a); end if: end do: ifactor(n);

n strong pseudoprime base 2 n pseudoprime base 3 n pseudoprime base 5 n strong pseudoprime base 7 n pseudoprime base 11 n pseudoprime base 17

KigtSSFHNiI2IyIjOCIiIi1GJDYjIiNoRigtRiQ2IyIkKFJGKA==

restart: Testing a big number n for strong pseudoprimes to bases in pr Choose n and pr n:=2^1279-1:pr:=[2,3,5,7,11,13,17,19,23,29]: sp:="n spp base %a ": m:=n-1:q:=m: r:=0: while type(q,even) do q:=iquo(q,2):r:=r+1: end do: for a in pr do e:=a&^q mod n: if e=1 then printf(sp,a); end if: for i from 0 to r-1 do if e=m then printf(sp,a); end if: e:=e*e mod n: end do; end do:

n spp base 2 n spp base 3 n spp base 5 n spp base 7 n spp base 11 n spp base 13 n spp base 17 n spp base 19 n spp base 23 n spp base 29

As you have discovered from the programs, MAPLE is well suited for (among others) applications like these.

\317\200 primes \317\200(x) primes less than x pp pseudoprime JkkjcHBHNiI2I0kiYUdGJA== pseudoprime to base a LUkiKkclKnByb3RlY3RlZEc2JCowJkkjcHBHNiI2I0kiYUdGKSIiIkkieEdGKUYsSSxwc2V1ZG9wcmltZUdGKUYsSSViYXNlR0YpRixGK0YsSSVsZXNzR0YpRixJJXRoYW5HRilGLEYt spp strong pseudoprime 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 KjItJkkkc3BwRzYiNiNJImFHRiY2I0kieEdGJiIiIkknc3Ryb25nR0YmRitJLHBzZXVkb3ByaW1lR0YmRitJJWJhc2VHRiZGK0YoRitJJWxlc3NHRiZGK0kldGhhbkdGJkYrRipGKw== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn Cmn Carmichael number Cmn(x) Carmichael number less than x \342\210\200a : 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 MC1JKmludGVyc2VjdEclKnByb3RlY3RlZEc2JCZJJHNwcEc2IjYjIiIjSSRDbW5HRik8Ig== Least element in each region \317\200 2 JkkjcHBHNiI2IyIiIw== 341 Jkkkc3BwRzYiNiMiIiM= 2047 Cmn 561 Cmn\342\213\202Jkkkc3BwRzYiNiMiIiM= 15841 Number of elements x = LUkiXkclKnByb3RlY3RlZEc2JCIjNSIiJw== \317\200(x) 78498 KiYmSSNwcEc2IjYjIiIjIiIiSSJ4R0YlRig= 245 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn KiYmSSRzcHBHNiI2IyIiIyIiIkkieEdGJUYo 46 Cmn(x) 43 (of these 23 with 3 factors, 19 with 4 factors, 1 with 5 factors)

1729 = 7*13*19 = 19*91 = LUkiXkclKnByb3RlY3RlZEc2JCIiIiIiJA==+ Ly1JIl5HJSpwcm90ZWN0ZWRHNiQiIzciIiQsJi1GJDYkIiIqRigiIiItRiQ2JCIjNUYoRi0=LyIlRzwtSSIrRyUqcHJvdGVjdGVkRzYkIiRIKCIlKzU=NiI=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 41041 = 7*11*13*41 = 1001*41 410041= 41*73*137 = 10001*41 101101 = 7*11*13*101 = 101*1001

Legal Notice: The copyright for this application is owned by the author(s). Neither Maplesoft nor the author are responsible for any errors contained within and are not liable for any damages resulting from the use of this material. This application is intended for non-commercial, non-profit use only. Contact the author for permission if you wish to use this application in for-profit activities. 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