<Text-field style="Heading 1" layout="Heading 1">Racine carr\303\251e modulo p premier impair.</Text-field>

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

Racine carr\303\251e modulo un entier.

restart;

with(numtheory);

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

<Text-field style="Heading 1" layout="Heading 1">Racine carr\303\251e modulo p premier impair.</Text-field>

Soit p un nombre premier impair. La proc\303\251dure suivante rend q impair et i tels que p-1=q.2^i.

impair:=proc(p) local i,q; i:=0; q:=p-1; while q mod 2 = 0 do i:=i+1; q:=q/2; od; [q,i]; end;

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impair(7);

impair(13);

Soit p un nombre premier impair tel que q est le plus grand facteur impair de p-1. La proc\303\251dure gene rend un g\303\251n\303\251rateur du 2-Sylow de (Z/pZ)^*.

gene:=proc(p,q) local n,h,z; h:=rand(2..p-1); n:=h(); while legendre(n,p)=1 do n:=h() od; z:=n&^q mod p; end;

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Soit p un nombre premier impair et a un entier premier \303\240 p. La proc\303\251dure racine rend une racine carr\303\251e de a modulo p.

racine:=proc(p,a) local x,y,z,q,e,r,b,v,c,t,m; v := impair(p) ; q := v[1]; e := v[2]; y := gene(p,q); r := e; x := a&^((q-1)/2) mod p; b := a*x^2 mod p; x := a*x mod p; while b mod p <> 1 do m := 1; c := b^2 mod p; while c mod p <> 1 do c := c^2 mod p; m := m+1; od; if m=r then RETURN('Erreur') fi; t := y&^(2^(r-m-1)) mod p; y := t^2 mod p; r := m; x := x*t mod p; b := b*y mod p; od; x; end;

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

p:=ithprime(100);

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

legendre(5,p);

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

racine(p,5);

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

345^2 mod p;

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

legendre(6,p);

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

racine(p,6);

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

<Text-field style="Heading 1" layout="Heading 1">Racines carr\303\251es modulo p^n (p premier impair).</Text-field>

racinepn:=proc(p,n,a) local x,t,q,i; x := racine(p,a); q := p; for i from 1 to n-1 do t := -(x^2-a)/q; t := t*(2*x)&^(-1) mod p; x := x+t*q mod q*p; q := q*p; od; [x,-x mod p^n]; end;

NiQtSSVtcm93RzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliR0YoNiUtSSNtaUdGJTY5USlyYWNpbmVwbkYoLyUnZmFtaWx5R1EuTHVjaWRhfkJyaWdodEYoLyUlc2l6ZUdRIzEyRigvJSVib2xkR1EmZmFsc2VGKC8lJ2l0YWxpY0dRJXRydWVGKC8lKnVuZGVybGluZUdGOC8lKnN1YnNjcmlwdEdGOC8lLHN1cGVyc2NyaXB0R0Y4LyUrZm9yZWdyb3VuZEdRKlswLDAsMjU1XUYoLyUrYmFja2dyb3VuZEdRKFswLDAsMF1GKC8lJ29wYXF1ZUdGOC8lK2V4ZWN1dGFibGVHRjgvJSlyZWFkb25seUdGOy8lKWNvbXBvc2VkR0Y4LyUqY29udmVydGVkR0Y4LyUraW1zZWxlY3RlZEdGOC8lLHBsYWNlaG9sZGVyR0Y4LyUwZm9udF9zdHlsZV9uYW1lR1EqMkR+T3V0cHV0RigvJSptYXRoY29sb3JHRkQvJS9tYXRoYmFja2dyb3VuZEdGRy8lK2ZvbnRmYW1pbHlHRjIvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YoLyUpbWF0aHNpemVHRjUtSSNtb0dGJTYzUSM6PUYoLyUlZm9ybUdRJmluZml4RigvJSZmZW5jZUdGOC8lKnNlcGFyYXRvckdGOC8lJ2xzcGFjZUdRL3RoaWNrbWF0aHNwYWNlRigvJSdyc3BhY2VHRltwLyUpc3RyZXRjaHlHRjgvJSpzeW1tZXRyaWNHRjgvJShtYXhzaXplR1EpaW5maW5pdHlGKC8lKG1pbnNpemVHUSIxRigvJShsYXJnZW9wR0Y4LyUubW92YWJsZWxpbWl0c0dGOC8lJ2FjY2VudEdGOC8lMGZvbnRfc3R5bGVfbmFtZUdGWC8lJXNpemVHRjUvJStmb3JlZ3JvdW5kR0ZELyUrYmFja2dyb3VuZEdGRy1GJDYpLUZfbzYzUSVwcm9jRigvRmNvUSFGKEZlb0Znby9Gam9RJDBlbUYoL0ZdcEZeckZecEZgcEZicEZlcEZocEZqcEZccUZecUZgcUZicUZkcS1GJDYlLUZfbzYzUSIoRigvRmNvUSdwcmVmaXhGKC9GZm9GO0Znby9Gam9RLnRoaW5tYXRoc3BhY2VGKC9GXXBGaXIvRl9wRjtGYHBGYnBGZXBGaHBGanBGXHFGXnFGYHFGYnFGZHEtRiQ2Jy1GLTY5USJwRihGMEYzRjZGOUY8Rj5GQEZCRkVGSEZKRkxGTkZQRlJGVEZWRllGZW5GZ25GaW5GXG8tRl9vNjNRIixGKEZib0Zlby9GaG9GO0Zdci9GXXBRM3Zlcnl0aGlja21hdGhzcGFjZUYoRl5wRmBwRmJwRmVwRmhwRmpwRlxxRl5xRmBxRmJxRmRxLUYtNjlRIm5GKEYwRjNGNkY5RjxGPkZARkJGRUZIRkpGTEZORlBGUkZURlZGWUZlbkZnbkZpbkZcb0Zhcy1GLTY5USJhRihGMEYzRjZGOUY8Rj5GQEZCRkVGSEZKRkxGTkZQRlJGVEZWRllGZW5GZ25GaW5GXG8tRl9vNjNRIilGKC9GY29RKHBvc3RmaXhGKEZnckZnb0Zoci9GXXBRMnZlcnl0aGlubWF0aHNwYWNlRihGW3NGYHBGYnBGZXBGaHBGanBGXHFGXnFGYHFGYnFGZHEtSSdtc3BhY2VHRiU2Ji8lJ2hlaWdodEdRJzAuMH5leEYoLyUmd2lkdGhHUScwLjN+ZW1GKC8lJmRlcHRoR0ZpdC8lKmxpbmVicmVha0dRMWZpcnN0cHJvY25ld2xpbmVGKC1GJDYmLUZfbzYzUSZsb2NhbEYoRltyRmVvRmdvRl1yL0ZdcFEwbWVkaXVtbWF0aHNwYWNlRihGXnBGYHBGYnBGZXBGaHBGanBGXHFGXnFGYHFGYnFGZHEtRiQ2KS1GLTY5USJ4RihGMEYzRjZGOUY8Rj5GQEZCRkVGSEZKRkxGTkZQRlJGVEZWRllGZW5GZ25GaW5GXG9GYXMtRi02OVEidEYoRjBGM0Y2RjlGPEY+RkBGQkZFRkhGSkZMRk5GUEZSRlRGVkZZRmVuRmduRmluRlxvRmFzLUYtNjlRInFGKEYwRjNGNkY5RjxGPkZARkJGRUZIRkpGTEZORlBGUkZURlZGWUZlbkZnbkZpbkZcb0Zhcy1GLTY5USJpRihGMEYzRjZGOUY8Rj5GQEZCRkVGSEZKRkxGTkZQRlJGVEZWRllGZW5GZ25GaW5GXG8tRl9vNjNRIjtGKEZib0Zlb0Zkc0ZdckZccEZecEZgcEZicEZlcEZocEZqcEZccUZecUZgcUZicUZkcUZkdC1GJDYmLUYkNiUtRiQ2JUZbdkZeby1GJDYlLUYtNjlRJ3JhY2luZUYoRjBGM0Y2RjlGPEY+RkBGQkZFRkhGSkZMRk5GUEZSRlRGVkZZRmVuRmduRmluRlxvLUZfbzYzUTAmQXBwbHlGdW5jdGlvbjtGKEZib0Zlb0Znb0ZdckZfckZecEZgcEZicEZlcEZocEZqcEZccUZecUZgcUZicUZkcS1GJDYlRmJyLUYkNiVGXnNGYXNGanNGXXRGZ3ZGZHQtRiQ2JS1GJDYlRmF2Rl5vRl5zRmd2RmR0LUYkNiUtRiQ2Jy1GJDYnLUZfbzYzUSRmb3JGKEZbckZlb0Znb0ZdckZndUZecEZgcEZicEZlcEZocEZqcEZccUZecUZgcUZicUZkcUZkdi1GX282M1EjdG9GKEZbckZlb0Znby9Gam9GaHVGZ3VGXnBGYHBGYnBGZXBGaHBGanBGXHFGXnFGYHFGYnFGZHEtRiQ2JUZncy1GX282M1EqfiZtaW51czt+RihGW3JGZW9GZ29GXXJGX3JGXnBGYHBGYnBGZXBGaHBGanBGXHFGXnFGYHFGYnFGZHEtSSNtbkdGJTY5RmdwRjBGM0Y2L0Y6RjhGPEY+RkBGQkZFRkhGSkZMRk5GUEZSRlRGVkZZRmVuRmduL0ZqblEnbm9ybWFsRihGXG8tRl9vNjNRI2RvRihGW3JGZW9GZ29GXHlGZ3VGXnBGYHBGYnBGZXBGaHBGanBGXHFGXnFGYHFGYnFGZHEtRmV0NiZGZ3RGanRGXXUvRmB1UTZpbmNyZWFzZWluZGVudG5ld2xpbmVGKC1GJDYmLUYkNiUtRiQ2JUZedkZeby1GJDYkLUZfbzYzUSgmbWludXM7RihGW3JGZW9GZ29GXXJGX3JGXnBGYHBGYnBGZXBGaHBGanBGXHFGXnFGYHFGYnFGZHEtRiQ2JS1GJDYlRmJyLUYkNiMtRiQ2JS1GJDYjLUYkNiVGW3YtRl9vNjNRIl5GKEZib0Zlb0Znby9Gam9GY3RGYnRGXnBGYHBGYnBGZXBGaHBGanBGXHFGXnFGYHFGYnFGZHEtRmN5NjlRIjJGKEYwRjNGNkZleUY8Rj5GQEZCRkVGSEZKRkxGTkZQRlJGVEZWRllGZW5GZ25GZnlGXG9GX3lGanNGXXQtRl9vNjNRIi9GKEZib0Zlb0Znb0ZockZqckZbc0ZgcEZicEZlcEZocEZqcEZccUZecUZgcUZicUZkcS1GJDYjRmF2Rmd2RmR0LUYkNiUtRiQ2JUZedkZeby1GJDYlLUYtNjlRJG1vZEYoRjBGM0Y2RjlGPEY+RkBGQkZFRkhGSkZMRk5GUEZSRlRGVkZZRmVuRmduRmluRlxvRmV3LUYkNiVGYnItRiQ2JS1GJDYlRl52LUZfbzYzUSIqRihGYm9GZW9GZ29GaHJGanJGXnBGYHBGYnBGZXBGaHBGanBGXHFGXnFGYHFGYnFGZHEtRiQ2JS1GJDYlRmJyLUYkNiVGaltsRmFdbEZbdkZddC1GX282M1ElfiZefkYoRltyRmVvRmdvRl1yRl9yRl5wRmBwRmJwRmVwRmhwRmpwRlxxRl5xRmBxRmJxRmRxLUYkNiVGYnItRmN5NjlRKSZtaW51czsxRihGMEYzRjZGZXlGPEY+RkBGQkZFRkhGSkZMRk5GUEZSRlRGVkZZRmVuRmduRmZ5RlxvRl10RmFzRl5zRl10Rmd2RmR0LUYkNiUtRiQ2JUZbdkZeby1GJDYlRmhcbEZldy1GJDYlRmJyLUYkNiUtRiQ2JUZbdi1GX282M1EiK0YoRmJvRmVvRmdvRlx5Rmd1Rl5wRmBwRmJwRmVwRmhwRmpwRlxxRl5xRmBxRmJxRmRxLUYkNiVGXnZGYV1sRmF2RmFzLUYkNiVGYXZGYV1sRl5zRl10Rmd2RmR0LUYkNiVGYXZGXm9GY19sLUZldDYmRmd0L0ZbdVEnMC4wfmVtRihGXXUvRmB1UTZkZWNyZWFzZWluZGVudG5ld2xpbmVGKC1GX282M1EnZW5kfmRvRihGW3JGZW9GZ29GXHlGX3JGXnBGYHBGYnBGZXBGaHBGanBGXHFGXnFGYHFGYnFGZHFGZ3ZGZHQtRiQ2JS1GX282M1EiW0YoRmVyRmdyRmdvRmhyRmpyRltzRmBwRmJwRmVwRmhwRmpwRlxxRl5xRmBxRmJxRmRxLUYkNiVGW3ZGYXMtRiQ2JUZoXGxGZXctRiQ2JUZici1GJDYlLUYkNiRGZ3pGW3ZGYXMtRiQ2JUZec0ZmW2xGZ3NGXXQtRl9vNjNRIl1GKEZgdEZnckZnb0ZockZidEZbc0ZgcEZicEZlcEZocEZqcEZccUZecUZgcUZicUZkcUZnX2wtRl9vNjNRKWVuZH5wcm9jRihGW3JGZW9GZ29GXHlGX3JGXnBGYHBGYnBGZXBGaHBGanBGXHFGXnFGYHFGYnFGZHE3Iy1fRilJLG1wcmludHNsYXNoR0YoNiQ3Iz5JKXJhY2luZXBuR0YoZio2JUkicEdGKEkibkdGKEkiYUdGKDYmSSJ4R0YoSSJ0R0YoSSJxR0YoSSJpR0YoRihGKEMmPkZlYmwtSSdyYWNpbmVHRig2JEZhYmxGY2JsPkZnYmxGYWJsPyhGaGJsIiIiRmBjbCwmRmJibEZgY2xGYGNsISIiSSV0cnVlRyUqcHJvdGVjdGVkR0MmPkZmYmwsJComLCYqJClGZWJsIiIjRmBjbEZgY2xGY2JsRmJjbEZgY2xGZ2JsRmJjbEZiY2w+RmZibC1JJG1vZEdGKDYkKiZGZmJsRmBjbC1JIyZeR0YoNiQsJComRlxkbEZgY2xGZWJsRmBjbEZgY2xGYmNsRmBjbEZhYmw+RmVibC1GX2RsNiQsJkZlYmxGYGNsKiZGZmJsRmBjbEZnYmxGYGNsRmBjbComRmdibEZgY2xGYWJsRmBjbD5GZ2JsRlxlbDckRmVibC1GX2RsNiQsJEZlYmxGYmNsKUZhYmxGYmJsRihGKEYoNyNGX2Js

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

r:=racinepn(p,7,5);

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

p^7;

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

r[1]^2 mod p^7;

<Text-field style="Heading 1" layout="Heading 1">Racines carr\303\251es modulo 2^n.</Text-field>

racine2n:=proc(n,a) local x,i,q,t; q := 4; x := 3; for i from 3 to n-1 do t := -(x^2-a)/(2*q) mod 2; x := x+t*q mod 2*q; q := 2*q; od; [x,-x mod 2^n,x+2^(n-1) mod 2^n,-x+2^(n-1) mod 2^n]; end;

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

racine2n(6,41);

13^2 mod 2^6;

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

<Text-field style="Heading 1" layout="Heading 1">Racines carr\303\251es modulo p^n (p premier quelconque).</Text-field>

racinepuiss:=proc(p,n,a) if p=2 then racine2n(n,a) else racinepn(p,n,a) fi end;

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<Text-field style="Heading 1" layout="Heading 1">Racines carr\303\251es modulo un entier n quelconque.</Text-field>

racinecarree:=proc(n,a) local i,f,t,v,m,tw,w,r,k; f := ifactors(n)[2]; t := nops(f); v := [seq(racinepuiss(f[j][1],f[j][2],a),j=1..t)]; m := x->convert(x,list); w := map(m,v[1]); tw := nops(w); for i from 2 to t do w := [seq([op(w[j]),v[i][1]],j=1..tw),seq([op(w[j]),v[i][2]],j=1..tw)]; tw := 2*tw; od; r := [seq(chrem(w[j],[seq(f[l][1]^f[l][2],l=1..t)]),j=1..tw)]; end;

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

rac:=racinecarree(144,73);

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

for i from 1 to 8 do rac[i]^2 mod 144 od;

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

<Text-field style="Heading 1" layout="Heading 1">Racines d'un polyn\303\264me P modulo un nombre premier p.</Text-field>

J'emploie ici la m\303\251thode na\303\257ve qui consiste \303\240 essayer tous les entiers entre 0 et p.

racpol:=proc(f,p) local i,l; l := NULL; for i from 0 to p-1 do if subs(x=i,f) mod p = 0 then l:=l,i fi od; [l]; end;

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

P:=x^6+x+1;

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

p:=ithprime(100);

racpol(P,p);

subs(x=19,P);

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

% mod p;

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

Pp:=diff(P,x);

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subs(x=19,Pp) mod p;

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

<Text-field style="Heading 1" layout="Heading 1">Racines d'un polyn\303\264me P modulo p^n.</Text-field>

On suppose avoir trouv\303\251 r tel que P(r)=0 mod p. On suppose que P'(r) est non nul modulo p.

racpolpn:=proc(P,p,n,r) local t,q,i,Pp,s; Pp := diff(P,x); s := r; q := p; for i from 1 to n-1 do t := -subs(x=s,P)/q; t := t*subs(x=r,Pp)&^(-1) mod p; s := s+t*q mod q*p; q := q*p; od; s; end;

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

racpolpn(P,p,4,19);

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

subs(x=%,P) mod 541^4;

<Text-field style="Heading 1" layout="Heading 1">Fonctions de maple.</Text-field>

msolve(x^2-5,541);

NiQtSSVtcm93RzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliR0YoNiUtRiQ2JS1JI21vR0YlNjNRInxmckYoLyUlZm9ybUdRJ3ByZWZpeEYoLyUmZmVuY2VHUSV0cnVlRigvJSpzZXBhcmF0b3JHUSZmYWxzZUYoLyUnbHNwYWNlR1EudGhpbm1hdGhzcGFjZUYoLyUncnNwYWNlR0Y9LyUpc3RyZXRjaHlHRjcvJSpzeW1tZXRyaWNHRjovJShtYXhzaXplR1EpaW5maW5pdHlGKC8lKG1pbnNpemVHUSIxRigvJShsYXJnZW9wR0Y6LyUubW92YWJsZWxpbWl0c0dGOi8lJ2FjY2VudEdGOi8lMGZvbnRfc3R5bGVfbmFtZUdRKjJEfk91dHB1dEYoLyUlc2l6ZUdRIzEyRigvJStmb3JlZ3JvdW5kR1EqWzAsMCwyNTVdRigvJStiYWNrZ3JvdW5kR1EoWzAsMCwwXUYoLUYkNiMtRiQ2JS1JI21pR0YlNjlRInhGKC8lJ2ZhbWlseUdRLkx1Y2lkYX5CcmlnaHRGKC8lJXNpemVHRlUvJSVib2xkR0Y6LyUnaXRhbGljR0Y3LyUqdW5kZXJsaW5lR0Y6LyUqc3Vic2NyaXB0R0Y6LyUsc3VwZXJzY3JpcHRHRjovJStmb3JlZ3JvdW5kR0ZYLyUrYmFja2dyb3VuZEdGZW4vJSdvcGFxdWVHRjovJStleGVjdXRhYmxlR0Y6LyUpcmVhZG9ubHlHRjcvJSljb21wb3NlZEdGOi8lKmNvbnZlcnRlZEdGOi8lK2ltc2VsZWN0ZWRHRjovJSxwbGFjZWhvbGRlckdGOi8lMGZvbnRfc3R5bGVfbmFtZUdGUi8lKm1hdGhjb2xvckdGWC8lL21hdGhiYWNrZ3JvdW5kR0Zlbi8lK2ZvbnRmYW1pbHlHRmBvLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGKC8lKW1hdGhzaXplR0ZVLUYvNjNRIj1GKC9GM1EmaW5maXhGKC9GNkY6RjgvRjxRL3RoaWNrbWF0aHNwYWNlRigvRj9GY3IvRkFGOkZCRkRGR0ZKRkxGTkZQRlNGVkZZLUkjbW5HRiU2OVEkMzQ1RihGXm9GYW9GY28vRmZvRjpGZ29GaW9GW3BGXXBGX3BGYXBGY3BGZXBGZ3BGaXBGW3FGXXFGX3FGYXFGY3FGZXEvRmhxUSdub3JtYWxGKEZqcS1GLzYzUSJ8aHJGKC9GM1EocG9zdGZpeEYoRjVGOEY7L0Y/UTJ2ZXJ5dGhpbm1hdGhzcGFjZUYoRkBGQkZERkdGSkZMRk5GUEZTRlZGWS1GLzYzUSIsRihGX3JGYXIvRjlGNy9GPFEkMGVtRigvRj9RM3Zlcnl0aGlja21hdGhzcGFjZUYoRmVyRkJGREZHRkpGTEZORlBGU0ZWRlktRiQ2JUYuLUYkNiMtRiQ2JUZqbkZcci1GZ3I2OVEkMTk2RihGXm9GYW9GY29GanJGZ29GaW9GW3BGXXBGX3BGYXBGY3BGZXBGZ3BGaXBGW3FGXXFGX3FGYXFGY3FGZXFGW3NGanFGXXM3IzYkPCMvSSJ4R0YoIiRYJDwjL0ZpdCIkJz4=

345^2 mod 541;

msqrt(5,541);

msqrt(5,541^10);

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

primroot(541^10);

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

mlog(5,2,541^10);

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

2&^% mod 541^10; # "&^" impose de faire les calculs interm\303\251diaires # modulo 541^10.

mlog(5,2,7);

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

Donc on ne peut pas trouver y tel que 2^y=5 mod 7.

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