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数字的奇妙：素数

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Entries A to Z. 7 Q( k& l$ W' i/ `2 t, x1 r3 v1 X
abc conjecture. 1 l; u4 W+ Y. C: n. T
abundant number.
, m# E' h8 a/ x( v7 k' @AKS algorithm for primality testing.
( ?1 c* y! ]( w' f/ ], M+ N: ]aliquot sequences (sociable chains). 4 z0 Y$ F$ q A" g, X" J9 s. m
almost-primes. - _+ A: E" w( U1 l! ]
amicable numbers. 2 e3 T U, [7 @+ U# u4 u$ g6 H4 _
amicable curiosities. 3 c1 G/ E( ?5 L9 s
Andrica’s conjecture.
+ v( U! b& y* p- Larithmetic progressions, of primes.
* S/ ~ C) u& d4 fAurifeuillian factorization.
# u) I, v3 r0 S( a0 R4 iaverage prime. 0 b+ Q7 c& V! a8 i( t
Bang’s theorem.
' i* }. h( E: _0 s/ a" o2 E8 T& \/ @Bateman’s conjecture. 7 A( u O/ u% r& S6 g! y( g
Beal’s conjecture, and prize.
/ F( k/ H% E- d L% D. w FBenford’s law. ) s! p5 ?: _& W! l8 C
Bernoulli numbers. 5 N0 `% d$ j. p$ ^ O6 I9 O
Bernoulli number curiosities.
& w+ o7 |8 q3 d" UBertrand’s postulate. . m9 I7 q# B/ K1 F6 B6 o
Bonse’s inequality.
' _/ Z0 J/ o4 s& o: K( }6 I8 rBrier numbers.
3 B" |: R9 |( F: m9 jBrocard’s conjecture.
/ y' f& f* ~! [8 x7 a! XBrun’s constant.
4 w5 d% {6 o& ?3 y) y4 v1 ZBuss’s function. 9 M* A* }/ |, J5 H& j3 o8 b9 s* \
Carmichael numbers. 6 e$ S$ ]" T _9 l* l
Catalan’s conjecture.
7 b' B3 q2 z; P1 B) k& T _( mCatalan’s Mersenne conjecture.
$ u" c* K% T6 y* i, S: W, SChampernowne’s constant. + r4 r4 h @- _
champion numbers.
5 T, Z( f1 C, D) } [3 R1 iChinese remainder theorem.
+ G5 ~0 r1 K/ A5 n s X# y) Pcicadas and prime periods.
( u/ G7 `/ u/ ~7 x& L( Gcircle, prime.
; S# Y* \8 L" j3 i: w3 E) P: p% Bcircular prime.
# r' a1 H" i* \! Q) O4 f- `Clay prizes, the. ) {8 j" w+ H6 f7 Z3 k( g& B& Z
compositorial. 9 s9 B' a. A- C, T% n+ L& i+ [
concatenation of primes. 5 E. O7 b, U3 \$ s* s! } r
conjectures. 9 C- h6 P& X- {$ @2 Q. `: s& K8 v
consecutive integer sequence. 1 b+ Z4 `8 J6 t% y1 a
consecutive numbers.
/ p4 S% ?1 k/ c7 |# v4 uconsecutive primes, sums of.
9 y7 J1 R' G, VConway’s prime-producing machine.
# t8 v4 m! O5 |6 f$ qcousin primes.
/ J: z9 y- x' m7 o: PCullen primes. + P! c: e$ L9 B' P; E( {6 u# N* I
Cunningham project. 4 I, S; n- _7 l7 y* E
Cunningham chains. 6 k9 G- A' y" {3 S
decimals, recurring (periodic). ( l# |4 X: A/ R
the period of 1/13.
# G! X8 h5 k) v! D% E$ n) t% K% Ocyclic numbers.
, T0 N7 r& k% |0 w) H0 o) @Artin’s conjecture.
( m) a; z5 D" {- o1 w* e0 @* @the repunit connection. + G* `% h5 V- O- m( @
magic squares.
- W0 k- W' v3 M2 v2 _deficient number.
7 f3 H) ~+ E* y z$ q; G2 e( Ldeletable and truncatable primes. . @' f. M# O D9 {( _7 M3 }
Demlo numbers.
6 N! {- W0 r, {% `& m3 Hdescriptive primes.
2 q6 M# n3 L7 N* R/ W5 A1 z4 MDickson’s conjecture.
: i. D2 H, X; W1 x/ odigit properties.
/ |. q: I7 t0 g+ v) ?* PDiophantus (c. AD 200; d. 284). * R+ a6 j, |' i9 S; g
Dirichlet’s theorem and primes in arithmetic series. 1 b3 o7 h2 @% C) n! S
primes in polynomials. 5 S+ U% a% T/ l4 z9 r7 k
distributed computing. / t( u/ ?7 j" \$ J$ F' X ]
divisibility tests.
 O% y! j, @ k# Y- s# W6 K/ t$ cdivisors (factors).
8 a) P8 F) w5 K: Ihow many divisors? how big is d(n)?
; v" d4 f$ m( K' \. xrecord number of divisors. & ?0 `5 s' ~( ^" s$ ]( E$ O
curiosities of d(n).
. u$ x3 A3 c% R( ]& sdivisors and congruences.
* p! n7 g1 `. Q' H4 Nthe sum of divisors function. V: B2 s" i& |! q U1 S
the size of σ(n). 9 i/ I- p' d1 O7 T9 x
a recursive formula.
3 w2 f0 t, W, a- N2 {2 ^divisors and partitions.
) H1 J6 z! R0 Q. S7 G. Y5 j; S; N, Pcuriosities of σ(n). 9 {$ z: J7 y5 G% Y* H/ ^4 ^: Q
prime factors. " }2 {) Q. |, _2 v7 l3 ?
divisor curiosities.
; H- B1 T- U! \" t' Zeconomical numbers. ! V8 r0 o8 `8 g6 C
Electronic Frontier Foundation. - C. n6 j1 j# I, g( U
elliptic curve primality proving.
, `- i1 M ?& f1 n( O2 Nemirp. + o8 B7 \4 x# f% T3 h
Eratosthenes of Cyrene, the sieve of.
- n. t. u9 @7 p( g* e: H5 B1 B8 k4 qErd?s, Paul (1913–1996). % H; s: T* I: r+ v* C! Y# F( L2 v
his collaborators and Erd?s numbers. 0 L5 H. o. ^* Y/ N3 j0 @* q6 d
errors.
- K4 @9 H/ C; `# [" KEuclid (c. 330–270 BC).
$ _) p! q9 t8 _unique factorization.
! T }( i2 C2 h! h&Radic;2 is irrational.
3 J7 O8 ^' e2 A+ d9 S" H# z2 kEuclid and the infinity of primes.
% s& H* F: t7 H; yconsecutive composite numbers. / b5 T) ?% u- {8 G
primes of the form 4n +3.
. A5 H1 g( r1 V* V$ e5 A2 b! Aa recursive sequence. * ~; K k/ N- d. d1 u% c. ^
Euclid and the first perfect number. " z; N b1 K* ^) s) J6 d
Euclidean algorithm. 0 V( f- B) D+ Y t$ }1 d1 J
Euler, Leonhard (1707–1783).
6 q1 m# H: T* T+ w* dEuler’s convenient numbers. 8 V# b4 U2 A- y
the Basel problem.
) V: E/ n- I0 t+ D5 Q2 ZEuler’s constant.
' L! U# O* X0 ?Euler and the reciprocals of the primes. + q7 t% W" S9 {6 i! f
Euler’s totient (phi) function. ; e& z& ^( x; Q
Carmichael’s totient function conjecture. 9 x% Z% e! x) E8 v6 U+ @
curiosities of φ(n). ( W1 t9 f' u M: b2 y, z8 o
Euler’s quadratic. 6 Y d5 E; V$ W# g3 t' Z6 |9 n% o
the Lucky Numbers of Euler.
: k4 r* k0 c. p* c- J+ Ofactorial.
+ e2 }8 L2 E6 Nfactors of factorials.
5 j" ~5 _2 a# c0 T' wfactorial primes. ( l: D9 W% d" ^5 v
factorial sums.
3 s1 K2 g* R# Q3 Wfactorials, double, triple . . . . 6 h$ i* e8 l! H. |+ d0 m% k. U
factorization, methods of. 9 |% W7 ^ q& S( F- o% I
factors of particular forms.
$ w. M- ^+ b( i2 N/ A% |& o7 VFermat’s algorithm.
. n" N1 h4 C5 W+ YLegendre’s method.
- r7 K0 s3 l6 H1 |congruences and factorization. / \0 ?% d! s0 |
how difficult is it to factor large numbers? 6 K6 Z" m/ X& q; z( {$ I
quantum computation. 0 e9 a3 y4 z! x8 l
Feit-Thompson conjecture. ( E3 d5 @" U7 ^1 e
Fermat, Pierre de (1607–1665). 9 V3 ~; E/ x4 n: Q6 s+ N% ?* g
Fermat’s Little Theorem.
7 X# Z! @: F1 r& yFermat quotient. 9 _; J4 b/ j# S* {) P5 c
Fermat and primes of the form x2 + y2. ) |; G s) o! h9 a' g8 F6 c) U
Fermat’s conjecture, Fermat numbers, and Fermat primes.
- ?: q- g* A" x& v, AFermat factorization, from F5 to F30. . a) d0 D+ X, s3 h0 p' q: h6 f
Generalized Fermat numbers.
( `2 Z" H8 G6 w$ O. y- Q. WFermat’s Last Theorem. " W0 B! f, T- N# [/ M, @8 O% y: ? t
the first case of Fermat’s Last Theorem.
 F# `8 B* q, s* DWall-Sun-Sun primes.
( j; j# }8 Z& l) w9 kFermat-Catalan equation and conjecture.
5 b3 ^0 R a. l* Q0 ^$ I1 L) XFibonacci numbers.
) W# n. y, l3 o3 G7 kdivisibility properties.
; E- T2 d0 {( a& }5 G$ iFibonacci curiosities.
3 d9 L" \* P4 q6 ^' ]& g5 Z' Iédouard Lucas and the Fibonacci numbers. 2 W6 C2 }! c! ?* n) f
Fibonacci composite sequences. . Q2 y1 P- h4 |4 [( c/ D# J4 |
formulae for primes.
, S& O7 r; K& YFortunate numbers and Fortune’s conjecture. ' i5 }1 O0 g/ i6 j
gaps between primes and composite runs. 2 s- l* v9 ^' x7 z: C- a+ t
Gauss, Johann Carl Friedrich (1777–1855).
; |) u9 ^7 n6 q5 {$ z9 X5 [Gauss and the distribution of primes. $ K& \! U. N/ k h7 S
Gaussian primes. ! L+ u1 J% S& p
Gauss’s circle problem. & x9 ]0 b/ F: L- ?3 Q% ~
Gilbreath’s conjecture.
. j' U {- K, r2 GGIMPS—Great Internet Mersenne Prime Search. 9 }5 v% P# ?- P& T% U
Giuga’s conjecture.
. p6 ^7 \! U) ~6 t* _Giuga numbers. . d( R$ d* {9 b' G; ~$ w: c# a
Goldbach’s conjecture. 5 L$ Z+ S: y$ @$ i% `+ t* u
good primes.
3 D$ I3 G& ?- d& M' ^Grimm’s problem. 1 c- o) ]) d* ^' j# K/ l
Hardy, G. H. (1877–1947). ' j! M5 p: Z. N: M, V6 Q
Hardy-Littlewood conjectures.
 m+ B, Q( @! s# j5 Aheuristic reasoning.
% x$ ^- J$ T1 x# A! |a heuristic argument by George Pólya.
5 s6 l# G" E$ p" XHilbert’s 23 problems.
2 t6 V/ x9 H& e, P# Ohome prime.
2 k7 C' c% T$ Z& U9 Yhypothesis H. 1 v3 [ T2 i0 u" Z
illegal prime. }+ _; k4 V- `) v
inconsummate number.
- b* e) P, k7 O3 B( u/ linduction.
2 t3 N' M) @9 U6 P3 G: Fjumping champion.
5 {' t* c/ @$ zk-tuples conjecture, prime.
1 b5 }' E; W3 u" O; c- m. \; s- i6 pknots, prime and composite. * H, u5 L6 Z7 P( j
Landau, Edmund (1877–1938). ) d/ S+ }1 k! A! P# B
left-truncatable prime. + B5 }% `8 g. F7 v: V5 U
Legendre, A. M. (1752–1833).
, B2 N( }' j1 w% S2 q' I! I) K/ vLehmer, Derrick Norman (1867–1938). ( Y# g j9 m8 | ?+ k3 N& B
Lehmer, Derrick Henry (1905–1991). + n$ c6 ~1 {+ x E$ ?) ]; z/ P- B
Linnik’s constant.
# y& h! E6 c3 C+ z8 v+ ]( J! eLiouville, Joseph (1809–1882).
2 q( Q B4 q! M) C& aLittlewood’s theorem. 4 ~3 z4 D4 y2 l
the prime numbers race.
* C9 J( B( o9 |Lucas, édouard (1842–1891). - g B; \' R N7 _
the Lucas sequence. " O$ `* R) \. U9 ^2 ^/ Y( _ i
primality testing.
. l* d1 Y! I4 s6 D! C+ \4 i+ uLucas’s game of calculation.
3 x C2 g$ E2 e4 s# Athe Lucas-Lehmer test. ' B! ?8 `' p4 o% O. [% e/ t
lucky numbers. 8 w+ [6 g/ N, H! z" u
the number of lucky numbers and primes.
9 g. }! A/ g1 {4 ]& ~ E" |“random” primes.
7 R% U1 Q" v" o5 p" }7 Bmagic squares.
 W7 D1 f j: V4 N8 bMatijasevic and Hilbert’s 10th problem.
1 `- O/ g' D6 v4 U7 KMersenne numbers and Mersenne primes.
: |, e7 l) _+ b- N5 h7 I- ZMersenne numbers.
& R O4 j- b0 uhunting for Mersenne primes.
3 E# J# v9 R" v. u, p+ ithe coming of electronic computers.
$ H# n! f8 I- A5 |8 \) H( u3 kMersenne prime conjectures. & n( r% v. g/ R" V) }* C
the New Mersenne conjecture. 2 ]& G6 @# ]/ k/ L+ I
how many Mersenne primes? % @8 ]( r- x, E
Eberhart’s conjecture. 2 j1 d' l7 ]3 q2 i: a" \/ h* m
factors of Mersenne numbers. # J" V3 @% i7 G4 G7 c* M
Lucas-Lehmer test for Mersenne primes. , S( V/ Y! C; ^6 N5 j4 `
Mertens constant.
# D/ [( n' Z" f2 O+ c% |Mertens theorem. % i, `; C, I0 J* ]9 m
Mills’ theorem.
: P1 j$ n2 {) `% m3 [+ H$ Y. FWright’s theorem.
3 p/ H) |6 H' L3 Fmixed bag.
2 U' K+ B* E9 @) Q3 z7 `' Gmultiplication, fast.
( R) }1 ^$ C8 _+ bNiven numbers.
- F# Z! m7 T3 m3 [9 k4 D) todd numbers as p + 2a2.
, b2 @' c1 G$ R( B% o5 QOpperman’s conjecture.
0 \. W" N8 v( O& L$ A9 epalindromic primes. $ Q0 c5 g7 S2 V( ?3 u1 A+ n) E
pandigital primes. 4 |2 g9 d _8 |3 o" D: ~
Pascal’s ** and the binomial coefficients.
: H% c1 ^) r7 D+ ~4 U5 ZPascal’s ** and Sierpinski’s gasket. & N! S( i P! s8 E
Pascal ** curiosities.
+ G; q- l% x: U4 ypatents on prime numbers.
: ]' \: J% Y; d! M0 }* ?Pépin’s test for Fermat numbers. . Y$ f! {9 p, ^; o
perfect numbers. ~# X' J" e$ B- R9 v
odd perfect numbers. 8 S8 X: m3 b1 w3 g- `2 J* ]. y
perfect, multiply. " p! ~. T4 m2 i6 u' C% C
permutable primes. $ ~, u0 ~8 l4 a6 c5 f( J% Q
π, primes in the decimal expansion of. / U' z6 m) ?# t% h
Pocklington’s theorem. 3 J6 i3 [9 [5 q! f' Y# J: p
Polignac’s conjectures.
! Z4 r5 c' E8 W* X" C& zPolignac or obstinate numbers. " P6 S! l. `* j+ P
powerful numbers.
/ M, t- T7 ^/ |& I8 A2 Aprimality testing.
& n; V- p% |+ m$ Q2 X. o! Aprobabilistic methods.
% V4 u& V1 N0 B& F. c. Y8 L. Eprime number graph.
6 W" x& `: P3 m% k% H2 v: Gprime number theorem and the prime counting function.
 B/ A8 L! Q" m7 uhistory.
* u0 s/ x5 ?) lelementary proof.
 S4 f- I( V. j6 C4 L, N) Srecord calculations.
7 P3 b5 B& e7 N) d1 u( kestimating p(n).
2 `: P' n3 N7 H! @8 Scalculating p(n). + \1 }0 k- K2 D
a curiosity.
( L0 o" G! _* f/ |6 Nprime pretender. ; \. ~3 ?' b8 B+ c
primitive prime factor.
5 [/ ], c9 `( X3 t2 h6 L3 tprimitive roots. 0 V$ r7 w( h' S
Artin’s conjecture. 3 n, T6 o" ~7 C+ a: D' _
a curiosity. - a# F- x' |1 u) ]3 C9 X( v
primordial. 2 C8 _2 ], }. c0 ]; I3 D
primorial primes.
. a( N9 E) X3 E4 q) DProth’s theorem.
% W* H; ~; w) s1 G5 e9 l% _( ~- Ipseudoperfect numbers. ) N( Z* z- J* T6 c1 k
pseudoprimes.
# F0 ~ W: _2 Gbases and pseudoprimes.
 v' |" X6 r9 k5 C# j+ ?pseudoprimes, strong. , z( X- q; I' }0 Q3 T8 {
public key encryption.
1 _: P1 m9 S* a8 q+ A* ^pyramid, prime.
: e4 Z7 _% H. P; \/ x" `4 C+ t* fPythagorean **s, prime. ! f9 Q6 q/ z! T* z
quadratic residues. " I/ n' {8 A' x4 s( T& c
residual curiosities.
, n# f: K; G6 m- Tpolynomial congruences. ( Z5 C- |6 M0 ]5 e, m0 k
quadratic reciprocity, law of. # H" U3 y: n4 j5 l5 D# H; K' |3 i
Euler’s criterion. ; ~ R& w2 T8 C; Z+ r
Ramanujan, Srinivasa (1887–1920). ) c3 o; G$ Q ~' O2 |; t( I& _4 s
highly composite numbers. % _/ Z9 Z2 v" A7 {
randomness, of primes. 0 @& B+ C: B" A" M
Von Sternach and a prime random walk. 7 Q' W1 H9 V7 O0 [; w; F
record primes. * |1 b. e0 f8 ~: e
some records. 3 L9 x0 w+ g& ^4 q( i8 M2 @' O
repunits, prime. , O( [" H- z7 X6 R* ^) t1 Z: v
Rhonda numbers. 4 d+ a1 A1 n* O( o
Riemann hypothesis.
, [3 _) o/ S K! p1 d( ~4 sthe Farey sequence and the Riemann hypothesis.
8 d$ `. O# K0 d3 Xthe Riemann hypothesis and σ(n), the sum of divisors function. 0 P! q+ X: m* s5 R! T% K
squarefree and blue and red numbers. 9 W1 x6 f$ G, i. B3 _/ }1 t9 A0 V
the Mertens conjecture. ! c6 {( s% p$ q& z. [9 V; ]
Riemann hypothesis curiosities.
' n, ]+ L" Z) R5 d' k+ p. BRiesel number. $ \" \! ^2 S" Q" L+ \8 c/ h5 d
right-truncatable prime. B* K/ l7 W' }# M0 r) L( s
RSA algorithm.
8 V. o3 l3 ^7 T& FMartin Gardner’s challenge.
% @% n5 n1 C% o* z- f4 b. dRSA Factoring Challenge, the New.
' k& T% x1 d; r! C6 C' j% I7 aRuth-Aaron numbers.
 l [" ^( j1 n9 m8 i2 s8 IScherk’s conjecture.
, ?% w; y- e I% dsemi-primes.
* @7 X# S) ]! w, @) {7 l. w**y primes. , S) F; y1 @% H9 B: d
Shank’s conjecture.
 ^- A v! V5 ISiamese primes. ! o1 Q8 E" O7 M6 E
Sierpinski numbers. , \* R8 ^! h D/ v( c6 u
Sierpinski strings.
: e, b6 i2 M, M4 Q- BSierpinski’s quadratic. 3 I' o! d5 R i5 A* G
Sierpinski’s φ(n) conjecture. 3 j8 B( y* B# r4 E( `, m: P
Sloane’s On-Line Encyclopedia of Integer Sequences.
7 \5 j. u9 T! E1 y9 _Smith numbers. Y4 g- I# I& \ N+ n% o1 q
Smith brothers. 1 J) p* }! Z& d. D3 I
smooth numbers. & J1 s7 L5 Q/ S4 [& W/ L
Sophie Germain primes.
- ~" m1 h0 r+ _/ Osafe primes. ; h* i. _; G- p |) H# l& X
squarefree numbers. - S; V0 m" V! ^( b6 w q- S+ K; s
Stern prime.
( y* k) b8 ~# @( Cstrong law of small numbers. & s1 X5 E% _4 z/ H
triangular numbers. 4 B1 G1 v. \6 {- E+ {
trivia. ; H1 r+ q# y; ]& c7 Q, { D- ]
twin primes.
$ i: [: s' n' Y0 k7 h3 qtwin curiosities.
* y$ z6 [, J2 c1 _3 m0 u& ]2 uUlam spiral.
, {2 M3 `/ X% {4 Tunitary divisors. , v$ p4 B0 U7 b/ ]# `
unitary perfect. : a' U+ P2 l0 V
untouchable numbers. ( A& d, W3 n9 s" w% A) [. z% }
weird numbers.
2 M3 \5 B, w, }Wieferich primes.
) g& Y' m/ k$ T1 s, Z! |9 [Wilson’s theorem.
9 c0 D: _. L/ i* xtwin primes. $ o3 w7 t7 O" {% \3 ^
Wilson primes.
1 D9 h2 J1 Y' SWolstenholme’s numbers, and theorems.
1 d3 b$ i4 W; L$ r! y# z' dmore factors of Wolstenholme numbers. ' G* \/ L% \4 U" L) L+ O
Woodall primes.
' Z1 w2 G: E3 w8 r+ nzeta mysteries: the quantum connection.