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数字的奇妙：素数

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Entries A to Z.
/ G }. x* D0 eabc conjecture.
& E! E7 v1 I5 ^4 g! ~1 h- gabundant number. ' P6 S, h4 j" f. g( B
AKS algorithm for primality testing.
; a9 C! r; u8 `aliquot sequences (sociable chains). 3 z$ M) E( x7 C' S% [, U! V: H
almost-primes.
! H# V4 r. g. T8 S4 S! s$ Tamicable numbers.
7 E3 F. r4 V8 Q* X0 v! w( Q$ {amicable curiosities.
; m J8 E/ {3 L: ^/ LAndrica’s conjecture.
8 G; t2 Z. ~1 y$ x6 O# P$ [arithmetic progressions, of primes. 0 s& b" z9 \( H
Aurifeuillian factorization.
' r8 r* b q S _9 _0 zaverage prime.
" A' K! B- [, K/ Y9 J! J2 O) nBang’s theorem. 7 y6 ~/ J a# e$ ?7 F3 V- I
Bateman’s conjecture.
0 S5 `- X) F' F2 c: dBeal’s conjecture, and prize. 7 [ D, o+ r' J& w
Benford’s law. 2 j! T: x3 ]2 Q( h* D' z
Bernoulli numbers. ; ]9 ?9 ^; U6 C% z0 H
Bernoulli number curiosities. 6 F8 E0 G- U8 |: W' n$ i) @
Bertrand’s postulate.
- y: W$ N) o4 h- QBonse’s inequality.
: |% v- Z- @ W" F2 H; M5 n8 U: a' pBrier numbers. + @0 K6 B4 J# G6 s
Brocard’s conjecture. + V& F/ ] L1 t0 z4 s
Brun’s constant. 6 _0 u5 E. b) W
Buss’s function. 7 f6 X( y6 [+ H ^2 ^ P* ~, [
Carmichael numbers. 3 w/ M$ C( {5 s0 v+ Q; g+ i
Catalan’s conjecture. ) r5 q R8 Y5 p8 ~1 B O1 ]
Catalan’s Mersenne conjecture.
' h1 C1 e$ W9 w# X# |! dChampernowne’s constant. + _9 \: J6 G; E% C5 ?# K, U
champion numbers. % d- x4 J& I& I" I
Chinese remainder theorem.
$ u K4 \! j! k; C( y; i2 B' B1 F1 ~cicadas and prime periods. / P9 c2 g' |) F( s
circle, prime.
/ H+ s( F }9 y5 `0 A/ kcircular prime. B' M) n% o# B0 f% C4 g1 A
Clay prizes, the. ; e% {* ]' x* V5 p
compositorial. 8 ~% k& ?4 n4 D9 X& N
concatenation of primes. ; X. }! J1 T7 ]2 y9 z! i& V# e
conjectures.
, c! l6 K4 o+ uconsecutive integer sequence. ; g" s+ p% P, Z" |( w
consecutive numbers. 2 y& f, w! L7 E* r
consecutive primes, sums of. 4 N; r' O* [% L3 V
Conway’s prime-producing machine.
; k+ u0 t. t1 [8 | {/ [cousin primes.
( I% g" {* C1 s. D& iCullen primes. 2 J( ^9 m) k _& y/ H; _, p% Z
Cunningham project. & ^' l/ S2 x9 b& q/ ^
Cunningham chains. 5 G9 g; B4 [' \! {0 a6 r/ s
decimals, recurring (periodic). ! U2 k8 e) ]8 `- z! F2 x" Z% D* r
the period of 1/13.
$ j0 X/ n$ K+ @* r/ ncyclic numbers. ! b4 d5 F2 ?: o% G7 x" [% \
Artin’s conjecture.
9 a' P; [$ _8 W+ Q; s0 t5 uthe repunit connection.
0 P0 H1 X1 V+ {) _. Hmagic squares. + e" n2 f+ I/ d- I: q
deficient number.
3 a4 s0 S( d* R% _6 m" cdeletable and truncatable primes.
4 O0 t( u% Y9 W' S) L& ?Demlo numbers. 3 [. Q9 \) L7 A2 v% V) f: |
descriptive primes. A" h. K" y( d% S& b* D
Dickson’s conjecture. 0 t0 q/ Y6 ~% X
digit properties. 3 j$ s! i" L7 D/ O: c3 z9 Q4 e
Diophantus (c. AD 200; d. 284). ) l" `3 [( z! V1 E) }' e' h
Dirichlet’s theorem and primes in arithmetic series. * R8 o4 i v7 i2 A* A4 D
primes in polynomials. % r( `$ g4 m3 u0 R
distributed computing.
" ?' H5 R, v! x4 @* u( Fdivisibility tests.
 Z: g6 `0 t X/ Tdivisors (factors). 7 R3 h' l! [+ ^/ `- c# T
how many divisors? how big is d(n)?
" c9 F* m5 M( Z6 x; Crecord number of divisors. ! q) M: p& I3 J
curiosities of d(n). 5 h$ I6 I" i4 B" a8 O
divisors and congruences. H: H5 a7 I# J
the sum of divisors function. # w/ a( U6 i0 ~ H0 Y% z3 |
the size of σ(n).
; i# n9 F0 Y+ A2 S- ~a recursive formula. $ w* u* @2 H# D. c, J; f7 ~3 b
divisors and partitions. ) Q0 y8 L+ Y7 z* Q: d
curiosities of σ(n). * q$ o5 p2 k" Z+ A! B9 y
prime factors. C0 N4 J* i3 w( E# X. }" o, A' Z
divisor curiosities. 3 f- {& ]/ ?; b4 i! b* s2 |
economical numbers.
- h! @0 o1 L! J% M: r2 [7 v8 k8 kElectronic Frontier Foundation.
* C0 ~8 h2 t, T7 [, B7 ^* K4 D/ G# Zelliptic curve primality proving.
5 |5 H' y. f9 Demirp.
& P8 H8 {- E6 w+ K# _5 N. x3 CEratosthenes of Cyrene, the sieve of. 6 T- u. g' b2 m% b* M! z
Erd?s, Paul (1913–1996). / t2 O% w6 ~2 {/ L& M7 b
his collaborators and Erd?s numbers.
( B3 {% G) S% n5 f2 K" I" Perrors. $ y6 h9 M( [( n3 M1 D& b
Euclid (c. 330–270 BC).
7 x5 k7 s6 E* k6 H/ e& ~3 Munique factorization.
% c3 {4 g! f8 i. Q9 E9 U&Radic;2 is irrational. % s& {9 g! h: n) |. Y) c
Euclid and the infinity of primes.
1 W ]) ~" q( ~$ `1 [) W! ?1 V6 F7 fconsecutive composite numbers. 1 z8 u- q+ c* s3 T7 l
primes of the form 4n +3. $ ^- A0 u; f5 s& O) v) O
a recursive sequence.
% I( B8 x0 D+ s2 P1 e" |Euclid and the first perfect number. ( a P9 W$ y! L
Euclidean algorithm. w7 \9 H( a$ F6 D+ I& d( m C
Euler, Leonhard (1707–1783).
( j/ d$ u3 Q# z' q9 \; R" d4 uEuler’s convenient numbers. 8 p7 U' W6 B( V% j
the Basel problem.
* O( G$ L" c2 Y8 \7 B/ ? S! g dEuler’s constant. x. B e- H, z n$ V
Euler and the reciprocals of the primes. 7 E9 x F) O0 R7 ?0 E
Euler’s totient (phi) function. 6 G7 K/ l; s% O3 Q0 a# u1 _/ {( x
Carmichael’s totient function conjecture.
7 h! g# P1 k0 w3 C1 ~% pcuriosities of φ(n).
: N6 D; R3 S0 uEuler’s quadratic.
- D: `) i7 \, U5 dthe Lucky Numbers of Euler. 3 x) L, `% X1 p/ `, p
factorial. 5 P* M( }5 N* m: [5 _ f
factors of factorials. ; [- D, O/ a1 \" J
factorial primes. 8 n/ W1 x1 Q- W1 R r
factorial sums.
5 O/ c0 H& Q1 T. J& Bfactorials, double, triple . . . . % }( M/ J4 r6 e, l. ~9 s+ i2 y
factorization, methods of. 0 ?6 h4 {& T, v9 a. U
factors of particular forms. + A0 ]0 i! h1 D: C8 I& f
Fermat’s algorithm. , H g( G$ G; ~5 ?6 L) t: x
Legendre’s method.
! M1 s4 ~/ z6 y( I" _# ]congruences and factorization. $ D6 v& J* B" Z2 s6 h! T5 o2 S% X
how difficult is it to factor large numbers?
# t3 z# L2 ]& Z1 E5 P; nquantum computation.
( u' R, x" H6 g5 CFeit-Thompson conjecture. , ^9 { X F) }
Fermat, Pierre de (1607–1665). ( ?& C! X' E6 S
Fermat’s Little Theorem.
" E7 V# H. i) L' e( B" ^Fermat quotient. ; ^8 ~- P! l2 L5 m' ^. h
Fermat and primes of the form x2 + y2. 2 r; g/ d% W, y6 R* i
Fermat’s conjecture, Fermat numbers, and Fermat primes.
8 B9 O1 {% q/ ]Fermat factorization, from F5 to F30.
! `3 e; M/ v1 j7 o7 ]* l% k, e& JGeneralized Fermat numbers. ( w9 H! {" S7 d2 F# q0 t4 C- |
Fermat’s Last Theorem. % }4 p6 q" z$ x l9 g& V5 J
the first case of Fermat’s Last Theorem.
$ I0 U; R7 \: @, P9 eWall-Sun-Sun primes. 8 ]* {' ~1 J7 W- \8 w" c
Fermat-Catalan equation and conjecture. 7 E" Z5 F J: T# v8 N
Fibonacci numbers. , Y+ M/ t; C8 H' |# s6 B+ Y
divisibility properties. * W# ~& P3 A4 Z6 y
Fibonacci curiosities.
3 E" Y( ?6 X0 i yédouard Lucas and the Fibonacci numbers. 7 c, q5 x0 L/ \# q
Fibonacci composite sequences.
1 {) j9 q* L- dformulae for primes.
8 U; R$ B; ~2 RFortunate numbers and Fortune’s conjecture. % Q% c6 W0 w! ]7 H
gaps between primes and composite runs.
0 ~& U# }% r. k! N/ G' @# X5 T; W7 YGauss, Johann Carl Friedrich (1777–1855).
+ k4 z% g$ t$ [+ |- v5 G, ^4 R: f* eGauss and the distribution of primes.
# d! E( I: K, v5 Y& n; C) xGaussian primes.
 c0 }1 V E% c( X C% j' b8 WGauss’s circle problem.
/ P! B0 c5 v/ o2 V+ c+ GGilbreath’s conjecture.
9 o$ L( _( @3 x4 g( J" T! eGIMPS—Great Internet Mersenne Prime Search.
% h: y" x. |$ ZGiuga’s conjecture.
6 X9 S- s, q; _6 p' h( WGiuga numbers. 0 r- G- X& ]( d* v) |! n; K' v
Goldbach’s conjecture. - b! N! {; P3 t9 e. E8 Y% M# I6 s
good primes.
6 p( u3 b" i; k1 e. d: T8 nGrimm’s problem.
0 c1 z1 @; A' @9 XHardy, G. H. (1877–1947). 7 G) p8 E. ?+ @: L0 [% P
Hardy-Littlewood conjectures.
; L5 @. \, h* F i( H' u- Jheuristic reasoning. - w7 y5 x+ n+ Y
a heuristic argument by George Pólya.
% n- }2 p% j3 O0 B K" D4 cHilbert’s 23 problems.
4 W2 E+ N* s6 t* r$ O5 r" E( Ohome prime.
; Q: E; v( I3 }hypothesis H.
: E4 [6 ?. j+ `# willegal prime.
/ ]5 K) u5 y6 z7 q5 Oinconsummate number.
# m5 x/ C9 f8 |+ pinduction.
$ E4 y& T3 _& Vjumping champion. ) g, O% E) A' u* U
k-tuples conjecture, prime. ' O% s+ x% R& N4 p H% Y! V' O% N- ^
knots, prime and composite.
( Q o+ p7 M8 a9 Q$ F9 l5 H# MLandau, Edmund (1877–1938). 5 h/ s+ G1 n$ z
left-truncatable prime. * e$ b- }8 l. f4 B
Legendre, A. M. (1752–1833). % Q. e+ o+ v T o
Lehmer, Derrick Norman (1867–1938). $ Q* M( ~' F5 @" r
Lehmer, Derrick Henry (1905–1991). 5 j s, K! s7 E4 x0 o
Linnik’s constant.
% J5 \5 P8 f' e) X+ m* `- k* |. {1 tLiouville, Joseph (1809–1882).
5 @7 b# z2 f6 T$ Y8 sLittlewood’s theorem. ! ]& B8 D, b" \5 o& j! @ _
the prime numbers race. ) y. |6 z1 L# F8 c
Lucas, édouard (1842–1891). ( J4 g) r! u6 U7 m" h2 x
the Lucas sequence. , g7 {+ {. F8 c. u% O
primality testing.
& O! i: @$ u2 \Lucas’s game of calculation. ( |( j: C( J- D, _) z" @) l% g
the Lucas-Lehmer test. g: _% o: }6 F& u6 j* T/ s
lucky numbers.
6 G! t( S! @+ @the number of lucky numbers and primes.
+ Y7 p- _3 F. k7 U" S+ f' q" ^“random” primes. : B E1 q3 k) p& q1 c4 d0 Q
magic squares.
' C& {3 E/ s( Y6 `+ XMatijasevic and Hilbert’s 10th problem.
* ~% ~7 W4 W' e% B, KMersenne numbers and Mersenne primes.
; m6 _) D/ B1 h3 J( MMersenne numbers.
% \0 V( ^5 C, l* \hunting for Mersenne primes.
" u6 t1 P$ n/ P: O/ E6 Ithe coming of electronic computers. 4 ^: a+ ~, M/ i. u
Mersenne prime conjectures.
+ C- V& D3 S3 b6 U4 p$ fthe New Mersenne conjecture.
0 x' d f8 q0 B$ e" ehow many Mersenne primes? " Z- b# t0 _6 V# n
Eberhart’s conjecture. " b6 ^- w' N6 [* t; n1 B, G
factors of Mersenne numbers. * r" A# X! u) Y0 J' Q3 e
Lucas-Lehmer test for Mersenne primes.
5 u6 ]) d5 y- Q, ?; \Mertens constant. 7 e. M; z+ f% X6 x9 W
Mertens theorem.
8 Z# W0 r' C+ h. ^( zMills’ theorem. " p3 G' O# k0 P) ^
Wright’s theorem.
, K' G5 N- f4 e" Q( w0 Xmixed bag. p( H2 t2 v0 |" `/ a
multiplication, fast.
) ~% v# [: D5 `1 {Niven numbers. / u& Z. S4 K4 v) P& h, s n
odd numbers as p + 2a2. ( L F7 l! N3 j: w8 y1 G$ J
Opperman’s conjecture. ' F3 ~* g/ C/ Z
palindromic primes. . F& I$ }1 ?3 e% b
pandigital primes.
4 D" Y9 K3 P( s8 u8 l. x8 ]Pascal’s ** and the binomial coefficients. " x7 N/ z. M1 o5 x7 _0 C. i$ Q
Pascal’s ** and Sierpinski’s gasket. 1 Z* D# Z2 R! X7 f
Pascal ** curiosities.
+ @1 C! P8 J# ]patents on prime numbers.
1 e& G9 q6 V. ~+ }5 _+ i5 YPépin’s test for Fermat numbers.
- g; `7 O# u1 O% _" tperfect numbers.
: N, a& |' B% Y6 o; y2 X( Nodd perfect numbers.
& A% A3 e3 _8 L9 N& Sperfect, multiply.
# ]* j* q$ K3 l6 T$ }7 @- ?permutable primes.
2 L& U1 [* g; W$ x9 V9 Wπ, primes in the decimal expansion of.
3 N: O& |( Z6 s" w, S+ lPocklington’s theorem. 9 F: b3 P! Q2 c3 ~
Polignac’s conjectures. 0 J- R" m! P' J/ r( K5 A* C
Polignac or obstinate numbers. - F+ r# u/ t$ Y: \$ ~
powerful numbers. J+ |$ G6 _; v: Z, ^( a
primality testing.
4 r, X6 C* I& I- B3 oprobabilistic methods.
7 ]6 t9 U5 b& v3 H9 ?prime number graph.
5 t( e; G+ E8 b% @) H' ~/ @; hprime number theorem and the prime counting function. * e9 ?% g' b; X+ A. D3 x" z8 D
history. & @: _% m0 I( d; B
elementary proof. ! q. H, f/ u, ^# g
record calculations. " z" D, _9 e; y% s2 v6 n8 P3 z* @" d6 Q
estimating p(n).
4 }. X& P6 ^2 \" g% |* z; Mcalculating p(n). 4 f6 P2 @ T3 R% s7 d
a curiosity. % P7 ^' X' r" r
prime pretender. - N f9 N% [& v( F9 `
primitive prime factor. , I1 c" y. o& v- }1 v
primitive roots.
2 `; o0 }) A" ? |+ tArtin’s conjecture. , o* }$ x8 L* `5 J6 l
a curiosity. 2 S) P; b* ?2 x* [3 U( l
primordial. , e6 h8 |+ ~ i- r, u% r+ `4 I8 G
primorial primes. ! {- S3 q6 Z: c, K( m
Proth’s theorem.
+ v f/ x5 [2 H8 l4 Hpseudoperfect numbers. & F6 s& O9 c$ [4 m; Y
pseudoprimes. % u: ^4 @7 u/ o i
bases and pseudoprimes. , l4 l9 S( P$ Q; Z2 D
pseudoprimes, strong. 6 |5 }6 C. q6 A4 X$ M+ r
public key encryption. 9 x U6 d0 O7 |- m, A1 O" A
pyramid, prime. : i9 q" r% t4 ?2 T7 X6 Q
Pythagorean **s, prime.
# W1 E7 \$ B5 k# I4 n$ equadratic residues.
 d: m2 l! q5 r6 @residual curiosities. 4 W6 I! w5 J M2 _5 @7 s
polynomial congruences.
# r& g, ^, ~5 k. \# h; Squadratic reciprocity, law of.
* B2 u: y, {; T8 W. v* VEuler’s criterion. : s7 I( @; L: q. D! c2 l
Ramanujan, Srinivasa (1887–1920). ' K: P% q+ S O4 F/ N V( \
highly composite numbers.
. i8 ^ h8 h% |4 m" u$ ?8 @randomness, of primes. , B" e% q- ]" B% y1 O) S' m# k# u
Von Sternach and a prime random walk. " G4 _" q) ^7 P1 _6 F6 ~
record primes.
) @5 K& R# O6 |8 Y; X0 \, `, j ~4 Msome records. 8 E. R' A# w- [ p: j' M
repunits, prime. $ o6 j- r/ U; }1 t
Rhonda numbers.
2 e3 e$ |$ u. C+ v6 {9 I+ n: CRiemann hypothesis. % V& A U+ N" N+ T
the Farey sequence and the Riemann hypothesis.
) c5 f* w! M6 Dthe Riemann hypothesis and σ(n), the sum of divisors function.
9 g2 T, e' z2 x: J- Vsquarefree and blue and red numbers.
7 n, ], R3 y% d6 [5 }! uthe Mertens conjecture. 3 s8 V9 h) {; R' |' b2 @
Riemann hypothesis curiosities. ( N+ W! `9 H6 m6 _ ^( i
Riesel number. - r% B; _ d3 p1 t( d3 a' \, W
right-truncatable prime.
. w' f% p9 d. X$ K1 _6 \3 h+ [1 tRSA algorithm. / }9 w1 l& L5 y V' k3 w! _
Martin Gardner’s challenge. " l" ~; ^ \- J" ~! _$ _& x0 O
RSA Factoring Challenge, the New. $ L' Y! ^" i4 Z2 i6 u
Ruth-Aaron numbers.
. q, C& f1 f6 e; u) E: q1 P1 yScherk’s conjecture. ) }8 |: \" b* o. S
semi-primes. 3 r3 p X, u) p6 Q; m% k
**y primes.
8 `' {# k% u9 Z; |9 x& d0 q, {Shank’s conjecture. ( y$ C2 O/ z' n
Siamese primes. . i; h( C( C3 `! }- x
Sierpinski numbers. 7 Z" \8 m5 K9 o* C' J) W/ F
Sierpinski strings. $ E$ k* |5 w& ?/ _+ h. [. S+ H
Sierpinski’s quadratic. 3 X" S8 H/ X, S( x, X6 u
Sierpinski’s φ(n) conjecture. * s; A$ S9 a& V6 @; v% j
Sloane’s On-Line Encyclopedia of Integer Sequences.
* P7 o) B' \$ sSmith numbers. 7 B; ^; W' x& G+ I' F1 O( v- W; @
Smith brothers. 8 p9 X# z4 l e, W% b& b4 g# Z6 I$ n
smooth numbers. - g4 N# l* C' S9 B
Sophie Germain primes.
: X* {9 x7 t+ |2 E. bsafe primes.
; b% |9 K8 E6 @9 Zsquarefree numbers.
) Q' l$ b- z2 N, V/ g7 vStern prime.
7 Z l, H3 u0 i* O5 ~& }0 Istrong law of small numbers. % O4 S; `( h6 Y5 W! A
triangular numbers. ( s4 P7 ]7 r- y7 M5 M
trivia.
( A# x, O' K! ?2 H# F/ h2 Utwin primes. % S' @1 ]' `9 S
twin curiosities. & _+ V0 k/ T' m# y) }4 |8 [9 K
Ulam spiral. + X/ E! b* S' M' D4 }# E
unitary divisors. 4 f% l3 [ B, M* |5 g( Q6 `. e
unitary perfect. 8 h- r6 ~8 C S
untouchable numbers.
+ S8 Z# Z% X6 a% h% V6 N5 q6 ~! Yweird numbers.
0 m( B6 C4 x" S( HWieferich primes. 9 l2 R+ }; U* { b& `- M5 D
Wilson’s theorem.
( @1 }0 o9 a* j2 Mtwin primes.
) K5 `) l* Y& {* s1 \8 j& DWilson primes. 9 Y: A5 m! l* u# L3 S% F
Wolstenholme’s numbers, and theorems. , s& F/ g2 ]/ w/ h" K
more factors of Wolstenholme numbers. & Q* P. R' p3 e9 C7 W k8 j, `+ f
Woodall primes.
3 a- g+ ?- U2 c e7 ^- u5 |zeta mysteries: the quantum connection.