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数字的奇妙：素数

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Entries A to Z.
- i Y8 @$ Z7 B8 p+ @! B% babc conjecture. ; H1 Y b5 T6 _) E
abundant number. & u7 w" U5 N s' k( l
AKS algorithm for primality testing. 1 G4 ]. \( x. D+ Z
aliquot sequences (sociable chains). & p; z# h- b! w! v/ ~
almost-primes.
7 C& C0 p$ h6 S0 A3 S/ w+ d: Jamicable numbers.
( {' v& w; K: ]- I7 damicable curiosities.
; J8 N: F# W0 a) l. i% H! ]Andrica’s conjecture. 3 y/ P0 R9 S, F3 Z/ V, A
arithmetic progressions, of primes.
, b- r' |; q( F5 H( qAurifeuillian factorization.
* `* ]9 }# a P5 S% {; ^0 Haverage prime.
) K4 f, N: c' ]8 u" i* i9 wBang’s theorem. 5 z; f3 @! g3 n( ]& O
Bateman’s conjecture. * l1 L, o) R f$ I0 F/ g8 F
Beal’s conjecture, and prize.
; Y/ U6 w4 T, B5 _2 P) p3 y4 V+ GBenford’s law.
% t+ A; u3 p* V9 ]+ R W1 {# F+ W, OBernoulli numbers. 1 l& y2 @# u/ a/ \* q% T
Bernoulli number curiosities.
4 H$ H' q9 u S3 R) P7 QBertrand’s postulate. ) f6 g9 W f( \2 l" \
Bonse’s inequality. . D/ s5 g$ y; R$ x
Brier numbers.
1 g2 u) j. v* J7 a5 I8 GBrocard’s conjecture.
8 [: |4 e8 o9 n8 |' i$ W$ ?Brun’s constant. : k- m0 V/ J. O7 W
Buss’s function.
- `) I6 ^ w- e; |Carmichael numbers. * A* {) `- z: o. C
Catalan’s conjecture.
0 ^ H% C i, SCatalan’s Mersenne conjecture. : a# v6 V* g ]( w( T, f5 @
Champernowne’s constant.
) ]0 ]- {- K5 Rchampion numbers. % h4 t) m# I; f; m
Chinese remainder theorem.
7 _. P" F! x* e. xcicadas and prime periods. + s$ _) m& Y/ S6 a% Y3 ]' P+ X
circle, prime.
0 R# [- P1 Y' J& \/ Qcircular prime. ! |. m8 X4 G2 Y ^0 L6 K, n
Clay prizes, the. 6 G# i+ B6 V, M* G4 S
compositorial. ( k$ |' C3 p* g0 `
concatenation of primes. , [; W8 d4 u3 i' X
conjectures.
0 w2 }2 r) K$ _, lconsecutive integer sequence. ( L: i, ~, u( E4 M6 ]
consecutive numbers. 4 x9 E) u9 f% M2 G/ |/ C% X
consecutive primes, sums of. : w2 p2 N7 A/ K, b$ y' H7 v3 s
Conway’s prime-producing machine.
) b/ |/ O( a% Ecousin primes.
+ L& ^5 T) u$ d/ X1 cCullen primes.
+ t6 S8 t) I1 [# r% ~* @Cunningham project. # C( z* L) {4 _& E7 D* ?
Cunningham chains.
- R. z8 c, u1 {) adecimals, recurring (periodic).
, o( Z; D0 Z" Dthe period of 1/13.
3 {& i. U" Q- a) F$ Zcyclic numbers.
3 X$ v; m% w+ o! x/ g" m' W! Y4 L) HArtin’s conjecture. ) J% d" N, d2 b8 A
the repunit connection. 7 I5 f5 A. F' J- `7 m6 P. }' x' d
magic squares. 2 @4 t8 }3 Y2 v& a$ |* S7 ~; N& f
deficient number.
5 Y0 L: }% [- h* r1 ?deletable and truncatable primes.
5 x$ V9 ^9 G. Y. d+ yDemlo numbers.
& p7 y8 t6 ?5 s, \& Z% Jdescriptive primes. 7 U5 R( b% C4 S( E$ H
Dickson’s conjecture.
( A, G8 ]- {% L7 m: R6 {digit properties.
4 j l) ~& q$ o6 V# }. gDiophantus (c. AD 200; d. 284). " k( l7 ~% R2 _/ D3 R; \. v
Dirichlet’s theorem and primes in arithmetic series. ' k- T1 T7 w7 N4 k$ n) q) X5 L0 s
primes in polynomials. # v4 [: j7 [4 l6 T1 f
distributed computing.
+ \; U6 w: V4 B1 R% g# G idivisibility tests. , r3 E6 c% @" K1 M/ {: h
divisors (factors). ( S& g8 s9 X8 P$ h$ m
how many divisors? how big is d(n)?
+ I, P: M8 c* ]: m2 Zrecord number of divisors.
; Y, k- {! `- {0 acuriosities of d(n). % X: @, r+ _, U5 h
divisors and congruences.
4 |% L' Z/ r- V9 w1 `+ _the sum of divisors function. 1 m. n# U! s( e3 l/ _* [
the size of σ(n). : N8 k% ~% R/ q5 v
a recursive formula.
! ^: X7 B: e8 x; M; k( i/ Ndivisors and partitions.
5 v' M }1 T9 U# s5 O' [curiosities of σ(n).
* f- _5 K/ J5 k2 uprime factors.
& g' C, V) t, E- E8 D9 W. edivisor curiosities. ! M+ W8 u/ l1 c5 j
economical numbers.
3 P, j6 q( s. m, ]$ rElectronic Frontier Foundation.
- h/ |1 x ~2 qelliptic curve primality proving. 5 ~1 \+ S2 P5 X( B4 o' g: K# x
emirp.
# ~9 p' \ h# i9 J: y% KEratosthenes of Cyrene, the sieve of. 9 O; g: W2 T/ w/ d6 A* U
Erd?s, Paul (1913–1996).
* n2 f8 Z4 J/ k% F% g; bhis collaborators and Erd?s numbers.
4 w: |& n/ L' @& C8 U. herrors. 6 Q, ?9 ]' G" Q8 X4 `4 e
Euclid (c. 330–270 BC). & a) U4 F2 E: T3 T1 N" w- n: O
unique factorization.
9 U6 A5 W4 W r( `4 S" _&Radic;2 is irrational. + |$ x/ e9 g6 R* p8 o! G
Euclid and the infinity of primes. $ m" S& l4 U2 R7 O
consecutive composite numbers. $ J6 W/ ~- B, y8 r% o' O5 c
primes of the form 4n +3. , `. N0 O# T) a u6 \
a recursive sequence. : G+ y4 y1 ?: C) ?- T' z: v
Euclid and the first perfect number. 1 B g% x& H% L' ~
Euclidean algorithm. ' x x) _$ t4 l& h. |. Z
Euler, Leonhard (1707–1783).
+ O& A' P$ D& Y$ j2 U6 BEuler’s convenient numbers. * n; v# W9 w" ]" |* R) Q
the Basel problem. 8 C$ V' o4 g) _2 ?2 K
Euler’s constant.
5 ]+ U2 i# G' x# d' @Euler and the reciprocals of the primes.
4 u* l3 v& h0 |" dEuler’s totient (phi) function.
% W. y3 k* p' s6 TCarmichael’s totient function conjecture.
4 ?; ^; w7 O' h! Wcuriosities of φ(n). + N1 g$ X; F. j* S0 X7 n
Euler’s quadratic.
7 K1 N& G' q; e+ I( [& zthe Lucky Numbers of Euler.
. [: H( y9 v& xfactorial. 7 n7 [2 \8 ]6 d5 w
factors of factorials. 7 Q) j1 y( r+ A H: y, n9 E
factorial primes.
, p( G# G& U' g' x5 }) ifactorial sums.
5 [6 x5 B; B% m* Efactorials, double, triple . . . . 8 q$ p' u% o/ D1 {% I; x
factorization, methods of.
4 ^* v8 c% g- Y! h! Wfactors of particular forms.
; y! {% Z$ i5 f/ E# nFermat’s algorithm.
- ]4 x, k+ N' x" z% GLegendre’s method. 5 J% K% M( m9 n2 p
congruences and factorization.
( Q4 U7 [2 J. r/ ?how difficult is it to factor large numbers?
/ P: T2 I. P7 x, ]$ J. q9 q1 Jquantum computation.
* o9 q9 c# Y6 h2 R) Q* @& JFeit-Thompson conjecture. $ |( l7 x! b' E, J) Z
Fermat, Pierre de (1607–1665). $ B" z# J; o' P# }
Fermat’s Little Theorem.
$ }2 ~' g8 s2 G. HFermat quotient. % N9 _6 h# Z t/ k$ R
Fermat and primes of the form x2 + y2.
# a; H! d# E5 R ^' SFermat’s conjecture, Fermat numbers, and Fermat primes.
& d- s! F5 Z3 E$ @/ _Fermat factorization, from F5 to F30. 7 \+ b) ^' s U8 m0 k# k8 w. O! q- Z, J
Generalized Fermat numbers. 7 V* r, k1 y) c9 k- M" N
Fermat’s Last Theorem. 2 M7 i: ?& {- {, \; Y o2 h5 h
the first case of Fermat’s Last Theorem. " S/ q. f' N6 w( K9 e, F) f; w* }
Wall-Sun-Sun primes. / B9 }* b- F m" c6 E
Fermat-Catalan equation and conjecture. 8 u4 W c2 W# _; f0 ^, I# A4 [
Fibonacci numbers.
 Y( J8 [( |0 y& G- _, S8 adivisibility properties. 6 B7 y+ \; O+ D0 \5 E4 G2 y: i
Fibonacci curiosities. ; Y+ H2 X2 P3 c& e. f, t' J
édouard Lucas and the Fibonacci numbers. / |# x$ p0 Y7 h
Fibonacci composite sequences.
- {- c; \: @" b2 s- zformulae for primes. 5 D0 U o- ]4 a% g5 I- e
Fortunate numbers and Fortune’s conjecture.
9 H; D) H' G( [% ?7 o1 A' x1 vgaps between primes and composite runs.
. V2 U* V Y1 [2 G( N" J! IGauss, Johann Carl Friedrich (1777–1855). : d' |* F* Y: n% `: { E
Gauss and the distribution of primes. ; {/ F" z. w, Z
Gaussian primes.
2 D$ Q! E' \1 C5 B* uGauss’s circle problem. ; g8 v* N9 f3 h" s q4 C8 k, \
Gilbreath’s conjecture.
% s4 x- G m- ]1 n3 K) d/ ^GIMPS—Great Internet Mersenne Prime Search. 4 ]' `9 \2 j# A z
Giuga’s conjecture. : d# q/ Q. p( H- V2 b( _
Giuga numbers.
- D" x" k/ O6 @- z0 t4 ~) CGoldbach’s conjecture.
1 q: {: Q* j lgood primes. $ n9 {2 E* m% R% N+ q# c
Grimm’s problem.
$ v' n7 }) c5 i; i9 s: K3 xHardy, G. H. (1877–1947). 1 @2 r! @2 e6 s2 ^' j% W0 N# G S
Hardy-Littlewood conjectures. ' V1 T6 }% P; e/ j
heuristic reasoning. 4 A: R$ a8 H) q( E. `
a heuristic argument by George Pólya. ' z( J; ?' Z* M
Hilbert’s 23 problems. 6 g6 Z# l, y8 j- D+ a e0 @4 }
home prime.
& f5 @# x% J5 n! C$ _% uhypothesis H. 3 p5 P: n. K6 D* r: }
illegal prime. ) ?. M/ B+ @$ p4 a6 l
inconsummate number. / _- _* E9 {6 T K7 b
induction.
) X' Z: g; |1 a% }2 S/ y9 ujumping champion. * b, C% j! W" p6 \8 j
k-tuples conjecture, prime. * r% L7 }# [+ I/ m* F' I
knots, prime and composite.
" m0 V4 b3 S9 w' {1 `Landau, Edmund (1877–1938). " n" U) e8 _. v( S0 i# o
left-truncatable prime.
. {8 N/ V+ \, T# Y9 Z* q: cLegendre, A. M. (1752–1833).
- E( e# ~. R! I5 a2 LLehmer, Derrick Norman (1867–1938). 8 j2 I3 s0 t1 l7 b/ E% `
Lehmer, Derrick Henry (1905–1991). . w7 a) A( N: O: O( _% ^
Linnik’s constant.
4 U, Q, P. Y# n+ I2 [Liouville, Joseph (1809–1882). ; U" g8 X, d- v: |
Littlewood’s theorem.
1 \: t$ t" U R0 F( g) h) x" uthe prime numbers race. 3 `6 ^+ m5 b0 \% U( p2 S
Lucas, édouard (1842–1891).
( f7 |) Y8 A x; \the Lucas sequence. 8 f' p3 P6 I, g& x; e4 x9 z" W
primality testing. 7 B) B @" \! I; a0 S% E+ H
Lucas’s game of calculation. , h+ N& C" W3 m8 U+ b6 l
the Lucas-Lehmer test. 1 Y- ?* a5 X: `8 S, ]- @
lucky numbers. 6 c0 V2 K' w i" j8 J
the number of lucky numbers and primes.
1 O4 ~8 I% ?5 v, W( o“random” primes.
- _2 }, |- T' J$ a3 m! G- ?3 bmagic squares. * x/ r1 p4 v$ z( q/ C/ b! [1 |( D- E9 ]
Matijasevic and Hilbert’s 10th problem.
3 u+ _, X, ~7 lMersenne numbers and Mersenne primes.
: X8 L2 ^0 y2 r8 Z' g* \Mersenne numbers. 9 C+ Q) q, H9 ~$ @ v
hunting for Mersenne primes.
" W( a! u2 V+ F6 {/ Cthe coming of electronic computers.
9 M Z6 M5 M- ?7 ~9 D3 [Mersenne prime conjectures. , |# z( w8 B \9 s5 n2 u! C
the New Mersenne conjecture.
! m$ `! z: r- h( Lhow many Mersenne primes?
) J& D% O7 h# `9 j( t6 k7 lEberhart’s conjecture. # j9 ?+ x. ?9 I- D% L
factors of Mersenne numbers.
: ?7 s, E9 T/ w8 N& VLucas-Lehmer test for Mersenne primes.
1 d+ X/ P- J: E" w1 hMertens constant. 0 O& f+ U( C2 N' Q7 d A
Mertens theorem.
1 l: n* f3 u8 [$ K% D- G: WMills’ theorem. , {0 l. w. g% k0 e: {' ]" C
Wright’s theorem. + p/ s% @6 Q2 j% F
mixed bag.
0 Z3 z+ ^$ k0 b& ]- y3 l2 lmultiplication, fast.
2 \- E) X% ~) W7 {Niven numbers. $ p5 S7 C K3 x3 l# S
odd numbers as p + 2a2.
% f# [1 k4 O N1 gOpperman’s conjecture. * t5 n8 G7 `( A
palindromic primes. 5 W& J: e0 Y; S4 K" M% B
pandigital primes. & v8 c5 I. E, x( n
Pascal’s ** and the binomial coefficients.
) _9 p: h v; F" }+ t( a" n8 yPascal’s ** and Sierpinski’s gasket.
/ d. L3 [4 l% R+ H; y# q, qPascal ** curiosities.
3 l6 A2 U. h* e3 Jpatents on prime numbers. 7 x+ P) T- J$ s
Pépin’s test for Fermat numbers.
$ d8 ? c* N; |: \. k! J$ T) j9 Qperfect numbers. * l6 U+ ]( A g- w: @
odd perfect numbers.
: D/ A8 b o. ], u; X0 f; [ \* gperfect, multiply.
" d+ C: r+ N+ v7 r3 x J, Mpermutable primes. 6 r$ K" A2 j3 `! h) J# D0 R+ Z; t# ?
π, primes in the decimal expansion of. & S A; y# L1 r- w( V, ~
Pocklington’s theorem.
; f) O: y u+ B) }8 x$ O3 pPolignac’s conjectures.
, z% r- ?6 `/ O1 c4 \( P- JPolignac or obstinate numbers.
* E% ~' ^% I' W8 Z" j+ v( Npowerful numbers. & Q$ H% `0 e* U9 H% j# |
primality testing.
; M4 }% l# O) r* V' S# b/ ^probabilistic methods. 6 l' c# B* s7 C3 C
prime number graph.
, Z3 ]9 k% S- Y. b; s/ Jprime number theorem and the prime counting function.
2 t$ `7 r* Z$ [" c0 Y9 ~history. 4 ^ H: a% x& _1 t8 m+ B
elementary proof.
7 p3 G% f# | h+ c8 I& Crecord calculations. 1 F# s m; G; r# u) ]3 Z( t1 H. A3 {
estimating p(n). 1 t2 s5 ?/ }5 A$ l% ]
calculating p(n).
: `2 y4 j8 t' c2 J1 m: c6 r% i, Ha curiosity. & A( u$ g* l" V+ |3 Q g
prime pretender.
( k. r+ N% T0 \+ v/ p! gprimitive prime factor. : X" s, j( d" g! z
primitive roots. 5 B; `. q& Y7 @% n4 w/ c
Artin’s conjecture. 5 v8 E: s' j; h' ^$ Y5 ^4 }& |( l
a curiosity. + d& j) V e5 Y4 M/ r1 N+ O
primordial. 6 b; ~. c7 k9 w# w9 d0 B
primorial primes.
6 E& `. C$ }% D" }# |Proth’s theorem.
7 E& B9 F& c. J; p% c5 S8 s) d% o, V. upseudoperfect numbers. 7 k" a, {' Q$ @, s, m
pseudoprimes. 3 j8 e; m# h( R% u$ u+ _' |
bases and pseudoprimes.
7 o5 {6 V: [% g/ ^4 T/ g' Q% Apseudoprimes, strong. + p1 z/ ~$ s, m
public key encryption. ( ]4 G7 V: d7 w8 _$ c
pyramid, prime. 1 p! v! W" O; _* ^
Pythagorean **s, prime.
' x% o7 A3 p; W0 B' zquadratic residues.
3 E. z3 j: b7 D- `5 w, S9 Nresidual curiosities.
/ o4 V: D: P m* H- \polynomial congruences.
4 o+ M2 G& P P) gquadratic reciprocity, law of. & `8 U) J) n/ f* A
Euler’s criterion.
% I$ G) Z! a" l6 ^2 Y0 k2 ARamanujan, Srinivasa (1887–1920).
6 X# b) e% F9 F* ~highly composite numbers.
& r ?5 _# C: v( p4 o# b5 srandomness, of primes. ( w4 }: S0 n0 T1 e* |
Von Sternach and a prime random walk. ( }% P" H) d3 ~6 C
record primes. ; V' t( I0 g' l r4 n7 U
some records.
' _! w# S2 b$ F' J+ @repunits, prime. 4 _( y6 x0 f; ^9 Q+ h* w+ D3 j7 z
Rhonda numbers. 9 W+ R1 N# l0 c" X
Riemann hypothesis.
7 A2 w4 S: F: n' P# lthe Farey sequence and the Riemann hypothesis.
9 k$ \ R& @, S% N) tthe Riemann hypothesis and σ(n), the sum of divisors function. + F- G2 @" ]& x& t7 N# y ?7 t0 f* d
squarefree and blue and red numbers. % P7 N) Y& U% O% P% f: y
the Mertens conjecture.
5 n% Y3 ~0 j! a% [2 P9 D) pRiemann hypothesis curiosities. / d; P( k% W @- ?2 ^7 L
Riesel number.
 ?( P U' q5 d6 X( r% Uright-truncatable prime.
2 L* |, G+ U* [/ @5 HRSA algorithm.
4 S7 x& ]7 t* ?0 eMartin Gardner’s challenge. 0 K& b3 u* b7 W* L( r
RSA Factoring Challenge, the New. - }) ~$ }! P7 X, n; o% g' J4 x4 M
Ruth-Aaron numbers. 7 n, s. S9 e' M! K" |9 b
Scherk’s conjecture. - T( C& Z9 ?. J4 f5 b+ V: X: ?+ ~9 A! u
semi-primes. 6 z9 x1 d) l) e0 D$ A: w3 G# {* K+ P
**y primes. : h$ X1 J7 Y$ G+ W; [4 G6 \
Shank’s conjecture.
5 L, g/ g. K4 Y4 r( N- TSiamese primes.
8 u: }8 D* [# A/ p3 s0 v7 L9 {% A7 OSierpinski numbers. + t/ I2 c% D- e( ~
Sierpinski strings.
7 I. k: u3 F8 @# p; G! R' s+ }1 }Sierpinski’s quadratic.
. ~3 S0 ?$ c6 u, R( lSierpinski’s φ(n) conjecture.
" g5 c5 ~- ^+ e6 M8 C; s1 ~7 MSloane’s On-Line Encyclopedia of Integer Sequences.
% E _; H. U6 B# z$ h- E6 t3 _) RSmith numbers. + E/ h9 ?8 m4 i
Smith brothers.
 E; C$ g$ P) jsmooth numbers.
% D$ _/ h( r* _+ K9 z: G& V7 ~+ {Sophie Germain primes. ( Z5 e K% W7 M% i" i7 E6 s! B
safe primes. + @0 [, b* N( `
squarefree numbers. 3 |. O0 b. ^# M) ] ?
Stern prime. 6 _' _8 d; Y$ m+ {# V1 g
strong law of small numbers.
7 u# S" d5 g' W, ntriangular numbers.
# \3 G% L6 T- D7 z) I" v1 Itrivia. - s \8 s; `3 \9 T% C# |4 q+ Z% A9 O
twin primes.
6 l/ T+ k) x5 ~* o3 e& jtwin curiosities.
/ N' C/ L2 k3 A0 EUlam spiral.
( \* @+ v6 E8 [. X: b; aunitary divisors. 8 ~% P' ^" Q {/ M1 R1 w; s
unitary perfect.
: ]; P" o; l/ ?, I! r' i3 [ Cuntouchable numbers. 3 f" s9 I* G/ V
weird numbers. : U1 s+ b5 u }9 M- E8 Q) Q
Wieferich primes. 9 S0 w* G7 U8 }) i, @5 b
Wilson’s theorem.
6 U, L7 ]7 |5 q, U( mtwin primes.
! z! {1 V3 v+ Z" Z/ s( T, [Wilson primes.
' p7 S* b# L! j9 X/ dWolstenholme’s numbers, and theorems.
% W' ]0 n8 Q6 Z* X0 Kmore factors of Wolstenholme numbers. & O6 a0 V' Z6 \' I1 i" b
Woodall primes.
. u) H1 h) P4 @2 o4 F: Hzeta mysteries: the quantum connection.