Prime numbers: a computational perspective
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York [u.a.]
Springer
2005
|
| Edition: | 2. ed. |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Auch als Internetausgabe |
| Physical Description: | XV, 597 S. |
| ISBN: | 9780387252827 |
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MARC
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| 100 | 1 | |a Crandall, Richard E. |d 1947-2012 |e Verfasser |0 (DE-588)11390732X |4 aut | |
| 245 | 1 | 0 | |a Prime numbers |b a computational perspective |c Richard Crandall ; Carl Pomerance |
| 250 | |a 2. ed. | ||
| 264 | 1 | |a New York [u.a.] |b Springer |c 2005 | |
| 300 | |a XV, 597 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 500 | |a Auch als Internetausgabe | ||
| 650 | 7 | |a Getallen |2 gtt | |
| 650 | 7 | |a Números primos |2 larpcal | |
| 650 | 7 | |a Teoria dos números |2 larpcal | |
| 650 | 4 | |a Numbers, Prime | |
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| 700 | 1 | |a Pomerance, Carl |d 1944- |e Verfasser |0 (DE-588)122920066 |4 aut | |
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| _version_ | 1804137753097535488 |
|---|---|
| adam_text | Contents
Preface
vu
1
PRIMES!
1
1.1 Problems
and progress
....................... 1
1.1.1
Fundamental theorem and fundamental problem
.... 1
1.1.2
Technological and algorithmic progress
.......... 2
1.1.3
The infinitude of primes
.................. 6
1.1.4
Asymptotic relations and order nomenclature
...... 8
1.1.5
How primes are distributed
................ 10
1.2
Celebrated conjectures and curiosities
.............. 14
1.2.1
Twin primes
........................ 14
1.2.2
Prime fc-tuples and hypothesis
H
............. 17
1.2.3
The
Goldbach
conjecture
................. 18
1.2.4
The convexity question
.................. 20
1.2.5
Prime-producing formulae
................. 21
1.3
Primes of special form
....................... 22
1.3.1
Mersenne primes
...................... 22
1.3.2
Fermat
numbers
...................... 27
1.3.3
Certain presumably rare primes
.............. 31
1.4
Analytic number theory
...................... 33
1.4.1
The Riemann
zeta
function
................ 33
1.4.2
Computational successes
.................. 38
1.4.3
Dirichlet ¿-functions
.................... 39
1.4.4
Exponential sums
...................... 43
1.4.5
Smooth numbers
...................... 48
1.5
Exercises
.............................. 49
1.6
Research problems
......................... 75
2
NUMBER-THEORETICAL TOOLS
83
2.1
Modular arithmetic
......................... 83
2.1.1
Greatest common divisor and inverse
........... 83
2.1.2
Powers
............................ 85
2.1.3
Chinese remainder theorem
................ 87
2.2
Polynomial arithmetic
....................... 89
2.2.1
Greatest common divisor for polynomials
........ 89
2.2.2
Finite fields
......................... 91
2.3
Squares and roots
.......................... 96
2.3.1
Quadratic residues
..................... 96
2.3.2
Square roots
........................ 99
2.3.3
Finding polynomial roots
................. 103
2.3.4
Representation by quadratic forms
............ 106
2.4
Exercises
.............................. 108
2.5
Research problems
......................... 113
RECOGNIZING PRIMES AND COMPOSITES
117
3.1
Trial division
............................117
3.1.1
Divisibility tests
......................117
3.1.2
Trial division
........................118
3.1.3
Practical considerations
..................119
3.1.4
Theoretical considerations
.................120
3.2
Sieving
................................121
3.2.1
Sieving to recognize primes
................121
3.2.2
Eratosthenes pseudocode
.................122
3.2.3
Sieving to construct a factor table
............122
3.2.4
Sieving to construct complete factorizations
.......123
3.2.5
Sieving to recognize smooth numbers
...........123
3.2.6
Sieving a polynomial
....................124
3.2.7
Theoretical considerations
.................126
3.3
Recognizing smooth numbers
...................128
3.4
Pseudoprimes
............................131
3.4.1
Fermat
pseudoprimes
...................131
3.4.2
Carmichael numbers
....................133
3.5
Probable primes and witnesses
..................135
3.5.1
The least witness for
η
...................140
3.6
Lucas
pseudoprimes
........................142
3.6.1
Fibonacci and Lucas
pseudoprimes
............142
3.6.2
Grantham s Frobenius test
................145
3.6.3
Implementing the Lucas and quadratic Frobenius tests
. 146
3.6.4
Theoretical considerations and stronger tests
......149
3.6.5
The general Frobenius test
................151
3.7
Counting primes
..........................152
3.7.1
Combinatorial method
...................152
3.7.2
Analytic method
......................158
3.8
Exercises
..............................162
3.9
Research problems
.........................168
PRIM
ALIT
Y
PROVING
173
4.1
The
η
- 1
test
...........................173
4.1.1
The Lucas theorem and
Pepin
test
............173
4.1.2
Partial factorization
....................174
4.1.3
Succinct certificates
....................179
4.2
The
», + 1
test
...........................181
4.2.1
The Lucas-Lehmer test
..................181
4.2.2
An improved
η
+ 1
test, and a combined n2
- 1
test
. . 184
4.2.3
Divisors in residue classes
................. 186
4.3
The finite field primality test
................... 190
4.4
Gauss and Jacobi sums
...................... 194
4.4.1
Gauss sums test
...................... 194
4.4.2
Jacobi sums test
...................... 199
4.5
The primality test of Agrawal, Kayal, and Saxena
(AKS
test)
. 200
4.5.1
Primality testing with roots of unity
........... 201
4.5.2
The complexity of Algorithm
4.5.1............ 205
4.5.3
Primality testing with Gaussian periods
......... 207
4.5.4
A quartic time primality test
............... 213
4.6
Exercises
.............................. 217
4.7
Research problems
......................... 222
EXPONENTIAL FACTORING ALGORITHMS
225
5.1
Squares
...............................225
5.1.1
Fermat
method
.......................225
5.1.2
Lehman method
......................227
5.1.3
Factor sieves
........................228
5.2
Monte Carlo methods
.......................229
5.2.1
Pollard rho method for factoring
.............229
5.2.2
Pollard rho method for discrete logarithms
.......232
5.2.3
Pollard lambda method for discrete logarithms
.....233
5.3
Baby-steps, giant-steps
.......................235
5.4
Pollard
ρ
- 1
method
........................236
5.5
Polynomial evaluation method
..................238
5.6
Binary quadratic forms
.......................239
5.6.1
Quadratic form fundamentals
...............239
5.6.2
Factoring with quadratic form representations
......242
5.6.3
Composition and the class group
.............245
5.6.4
Ambiguous forms and factorization
............248
5.7
Exercises
..............................251
5.8
Research problems
.........................255
SUBEXPONENTIAL FACTORING ALGORITHMS
261
6.1
The quadratic sieve factorization method
............261
6.1.1
Basic QS
..........................261
6.1.2
Basic QS: A summary
...................266
6.1.3
Fast matrix methods
....................268
6.1.4
Large prime variations
...................270
6.1.5
Multiple polynomials
....................273
6.1.6
Self initialization
......................274
6.1.7
Zhang s special quadratic sieve
..............276
6.2
Number field sieve
.........................278
6.2.1
Basic NFS: Strategy
....................279
6.2.2
Basic NFS: Exponent vectors
...............280
6.2.3
Basic
NFS: Complexity
.................. 285
6.2.4
Basic NFS: Obstructions
.................. 288
6.2.5
Basic NFS: Square roots
.................. 291
6.2.6
Basic NFS: Summary algorithm
.............. 292
6.2.7
NFS: Further considerations
................ 294
6.3
Rigorous factoring
......................... 301
6.4
Index-calculus method for discrete logarithms
.......... 302
6.4.1
Discrete logarithms in prime finite fields
......... 303
6.4.2
Discrete logarithms via smooth polynomials and smooth
algebraic integers
...................... 305
6.5
Exercises
.............................. 306
6.6
Research problems
......................... 315
ELLIPTIC CURVE ARITHMETIC
319
7.1
Elliptic curve fundamentals
....................319
7.2
Elliptic arithmetic
.........................323
7.3
The theorems of
Hasse, Deuring,
and Lenstra
..........333
7.4
Elliptic curve method
.......................335
7.4.1
Basic ECM algorithm
...................336
7.4.2
Optimization of ECM
...................339
7.5
Counting points on elliptic curves
.................347
7.5.1
Shanks-Mestre method
..................347
7.5.2
Schoof
method
.......................351
7.5.3
Atkin-Morain method
...................358
7.6
Elliptic curve primality proving (ECPP)
.............368
7.6.1
Goldwasser-Kilian primality test
.............368
7.6.2
Atkin-Morain primality test
................371
7.6.3
Fast primality-proving via ellpitic curves (fastECPP)
. 373
7.7
Exercises
..............................374
7.8
Research problems
.........................380
THE UBIQUITY OF PRIME NUMBERS
387
8.1
Cryptography
............................387
8.1.1
Diffie-Hellman key exchange
...............387
8.1.2
RSA cryptosystem
.....................389
8.1.3
Elliptic curve cryptosystems (ECCs)
...........391
8.1.4
Coin-flip protocol
......................396
8.2
Random-number generation
....................397
8.2.1
Modular methods
......................398
8.3
Quasi-Monte
Carlo (qMC) methods
...............404
8.3.1
Discrepancy theory
.....................404
8.3.2
Specific qMC sequences
..................407
8.3.3
Primes on Wall Street?
..................409
8.4
Diophantine analysis
........................415
8.5
Quantum computation
.......................418
8.5.1
Intuition on quantum Turing machines (QTMs)
.....419
8.5.2
The Shoř
quantum algorithm for factoring
........422
8.6
Curious, anecdotal, and interdisciplinary references to primes
. 424
8.7
Exercises
..............................431
8.8
Research problems
.........................436
9
PAST ALGORITHMS FOR LARGE-INTEGER
ARITHMETIC
443
9.1
Tour of grammar-school methods
................443
9.1.1
Multiplication
........................443
9.1.2
Squaring
...........................444
9.1.3
Div
and mod
........................445
9.2
Enhancements to modular arithmetic
...............447
9.2.1
Montgomery method
....................447
9.2.2
Newton methods
......................450
9.2.3
Moduli of special form
...................454
9.3
Exponentiation
...........................457
9.3.1
Basic binary ladders
....................458
9.3.2
Enhancements to ladders
.................460
9.4
Enhancements for gcd and inverse
................463
9.4.1
Binary gcd algorithms
...................463
9.4.2
Special inversion algorithms
................465
9.4.3
Recursive-gcd schemes for very large operands
.....466
9.5
Large-integer multiplication
....................473
9.5.1
Karatsuba and Toom-Cook methods
...........473
9.5.2
Fourier transform algorithms
...............476
9.5.3
Convolution theory
.....................488
9.5.4
Discrete weighted transform (DWT) methods
......493
9.5.5
Number-theoretical transform methods
..........498
9.5.6 Schönhage
method
.....................502
9.5.7
Nussbaumer method
....................503
9.5.8
Complexity of multiplication algorithms
.........506
9.5.9
Application to the Chinese remainder theorem
.....508
9.6
Polynomial arithmetic
.......................509
9.6.1
Polynomial multiplication
.................510
9.6.2
Fast polynomial inversion and remaindering
.......511
9.6.3
Polynomial evaluation
...................514
9.7
Exercises
..............................518
9.8
Research problems
.........................535
Appendix: BOOK PSEUDOCODE
541
References
547
Index
577
|
| adam_txt |
Contents
Preface
vu
1
PRIMES!
1
1.1 Problems
and progress
. 1
1.1.1
Fundamental theorem and fundamental problem
. 1
1.1.2
Technological and algorithmic progress
. 2
1.1.3
The infinitude of primes
. 6
1.1.4
Asymptotic relations and order nomenclature
. 8
1.1.5
How primes are distributed
. 10
1.2
Celebrated conjectures and curiosities
. 14
1.2.1
Twin primes
. 14
1.2.2
Prime fc-tuples and hypothesis
H
. 17
1.2.3
The
Goldbach
conjecture
. 18
1.2.4
The convexity question
. 20
1.2.5
Prime-producing formulae
. 21
1.3
Primes of special form
. 22
1.3.1
Mersenne primes
. 22
1.3.2
Fermat
numbers
. 27
1.3.3
Certain presumably rare primes
. 31
1.4
Analytic number theory
. 33
1.4.1
The Riemann
zeta
function
. 33
1.4.2
Computational successes
. 38
1.4.3
Dirichlet ¿-functions
. 39
1.4.4
Exponential sums
. 43
1.4.5
Smooth numbers
. 48
1.5
Exercises
. 49
1.6
Research problems
. 75
2
NUMBER-THEORETICAL TOOLS
83
2.1
Modular arithmetic
. 83
2.1.1
Greatest common divisor and inverse
. 83
2.1.2
Powers
. 85
2.1.3
Chinese remainder theorem
. 87
2.2
Polynomial arithmetic
. 89
2.2.1
Greatest common divisor for polynomials
. 89
2.2.2
Finite fields
. 91
2.3
Squares and roots
. 96
2.3.1
Quadratic residues
. 96
2.3.2
Square roots
. 99
2.3.3
Finding polynomial roots
. 103
2.3.4
Representation by quadratic forms
. 106
2.4
Exercises
. 108
2.5
Research problems
. 113
RECOGNIZING PRIMES AND COMPOSITES
117
3.1
Trial division
.117
3.1.1
Divisibility tests
.117
3.1.2
Trial division
.118
3.1.3
Practical considerations
.119
3.1.4
Theoretical considerations
.120
3.2
Sieving
.121
3.2.1
Sieving to recognize primes
.121
3.2.2
Eratosthenes pseudocode
.122
3.2.3
Sieving to construct a factor table
.122
3.2.4
Sieving to construct complete factorizations
.123
3.2.5
Sieving to recognize smooth numbers
.123
3.2.6
Sieving a polynomial
.124
3.2.7
Theoretical considerations
.126
3.3
Recognizing smooth numbers
.128
3.4
Pseudoprimes
.131
3.4.1
Fermat
pseudoprimes
.131
3.4.2
Carmichael numbers
.133
3.5
Probable primes and witnesses
.135
3.5.1
The least witness for
η
.140
3.6
Lucas
pseudoprimes
.142
3.6.1
Fibonacci and Lucas
pseudoprimes
.142
3.6.2
Grantham's Frobenius test
.145
3.6.3
Implementing the Lucas and quadratic Frobenius tests
. 146
3.6.4
Theoretical considerations and stronger tests
.149
3.6.5
The general Frobenius test
.151
3.7
Counting primes
.152
3.7.1
Combinatorial method
.152
3.7.2
Analytic method
.158
3.8
Exercises
.162
3.9
Research problems
.168
PRIM
ALIT
Y
PROVING
173
4.1
The
η
- 1
test
.173
4.1.1
The Lucas theorem and
Pepin
test
.173
4.1.2
Partial factorization
.174
4.1.3
Succinct certificates
.179
4.2
The
», + 1
test
.181
4.2.1
The Lucas-Lehmer test
.181
4.2.2
An improved
η
+ 1
test, and a combined n2
- 1
test
. . 184
4.2.3
Divisors in residue classes
. 186
4.3
The finite field primality test
. 190
4.4
Gauss and Jacobi sums
. 194
4.4.1
Gauss sums test
. 194
4.4.2
Jacobi sums test
. 199
4.5
The primality test of Agrawal, Kayal, and Saxena
(AKS
test)
. 200
4.5.1
Primality testing with roots of unity
. 201
4.5.2
The complexity of Algorithm
4.5.1. 205
4.5.3
Primality testing with Gaussian periods
. 207
4.5.4
A quartic time primality test
. 213
4.6
Exercises
. 217
4.7
Research problems
. 222
EXPONENTIAL FACTORING ALGORITHMS
225
5.1
Squares
.225
5.1.1
Fermat
method
.225
5.1.2
Lehman method
.227
5.1.3
Factor sieves
.228
5.2
Monte Carlo methods
.229
5.2.1
Pollard rho method for factoring
.229
5.2.2
Pollard rho method for discrete logarithms
.232
5.2.3
Pollard lambda method for discrete logarithms
.233
5.3
Baby-steps, giant-steps
.235
5.4
Pollard
ρ
- 1
method
.236
5.5
Polynomial evaluation method
.238
5.6
Binary quadratic forms
.239
5.6.1
Quadratic form fundamentals
.239
5.6.2
Factoring with quadratic form representations
.242
5.6.3
Composition and the class group
.245
5.6.4
Ambiguous forms and factorization
.248
5.7
Exercises
.251
5.8
Research problems
.255
SUBEXPONENTIAL FACTORING ALGORITHMS
261
6.1
The quadratic sieve factorization method
.261
6.1.1
Basic QS
.261
6.1.2
Basic QS: A summary
.266
6.1.3
Fast matrix methods
.268
6.1.4
Large prime variations
.270
6.1.5
Multiple polynomials
.273
6.1.6
Self initialization
.274
6.1.7
Zhang's special quadratic sieve
.276
6.2
Number field sieve
.278
6.2.1
Basic NFS: Strategy
.279
6.2.2
Basic NFS: Exponent vectors
.280
6.2.3
Basic
NFS: Complexity
. 285
6.2.4
Basic NFS: Obstructions
. 288
6.2.5
Basic NFS: Square roots
. 291
6.2.6
Basic NFS: Summary algorithm
. 292
6.2.7
NFS: Further considerations
. 294
6.3
Rigorous factoring
. 301
6.4
Index-calculus method for discrete logarithms
. 302
6.4.1
Discrete logarithms in prime finite fields
. 303
6.4.2
Discrete logarithms via smooth polynomials and smooth
algebraic integers
. 305
6.5
Exercises
. 306
6.6
Research problems
. 315
ELLIPTIC CURVE ARITHMETIC
319
7.1
Elliptic curve fundamentals
.319
7.2
Elliptic arithmetic
.323
7.3
The theorems of
Hasse, Deuring,
and Lenstra
.333
7.4
Elliptic curve method
.335
7.4.1
Basic ECM algorithm
.336
7.4.2
Optimization of ECM
.339
7.5
Counting points on elliptic curves
.347
7.5.1
Shanks-Mestre method
.347
7.5.2
Schoof
method
.351
7.5.3
Atkin-Morain method
.358
7.6
Elliptic curve primality proving (ECPP)
.368
7.6.1
Goldwasser-Kilian primality test
.368
7.6.2
Atkin-Morain primality test
.371
7.6.3
Fast primality-proving via ellpitic curves (fastECPP)
. 373
7.7
Exercises
.374
7.8
Research problems
.380
THE UBIQUITY OF PRIME NUMBERS
387
8.1
Cryptography
.387
8.1.1
Diffie-Hellman key exchange
.387
8.1.2
RSA cryptosystem
.389
8.1.3
Elliptic curve cryptosystems (ECCs)
.391
8.1.4
Coin-flip protocol
.396
8.2
Random-number generation
.397
8.2.1
Modular methods
.398
8.3
Quasi-Monte
Carlo (qMC) methods
.404
8.3.1
Discrepancy theory
.404
8.3.2
Specific qMC sequences
.407
8.3.3
Primes on Wall Street?
.409
8.4
Diophantine analysis
.415
8.5
Quantum computation
.418
8.5.1
Intuition on quantum Turing machines (QTMs)
.419
8.5.2
The Shoř
quantum algorithm for factoring
.422
8.6
Curious, anecdotal, and interdisciplinary references to primes
. 424
8.7
Exercises
.431
8.8
Research problems
.436
9
PAST ALGORITHMS FOR LARGE-INTEGER
ARITHMETIC
443
9.1
Tour of "grammar-school" methods
.443
9.1.1
Multiplication
.443
9.1.2
Squaring
.444
9.1.3
Div
and mod
.445
9.2
Enhancements to modular arithmetic
.447
9.2.1
Montgomery method
.447
9.2.2
Newton methods
.450
9.2.3
Moduli of special form
.454
9.3
Exponentiation
.457
9.3.1
Basic binary ladders
.458
9.3.2
Enhancements to ladders
.460
9.4
Enhancements for gcd and inverse
.463
9.4.1
Binary gcd algorithms
.463
9.4.2
Special inversion algorithms
.465
9.4.3
Recursive-gcd schemes for very large operands
.466
9.5
Large-integer multiplication
.473
9.5.1
Karatsuba and Toom-Cook methods
.473
9.5.2
Fourier transform algorithms
.476
9.5.3
Convolution theory
.488
9.5.4
Discrete weighted transform (DWT) methods
.493
9.5.5
Number-theoretical transform methods
.498
9.5.6 Schönhage
method
.502
9.5.7
Nussbaumer method
.503
9.5.8
Complexity of multiplication algorithms
.506
9.5.9
Application to the Chinese remainder theorem
.508
9.6
Polynomial arithmetic
.509
9.6.1
Polynomial multiplication
.510
9.6.2
Fast polynomial inversion and remaindering
.511
9.6.3
Polynomial evaluation
.514
9.7
Exercises
.518
9.8
Research problems
.535
Appendix: BOOK PSEUDOCODE
541
References
547
Index
577 |
| any_adam_object | 1 |
| any_adam_object_boolean | 1 |
| author | Crandall, Richard E. 1947-2012 Pomerance, Carl 1944- |
| author_GND | (DE-588)11390732X (DE-588)122920066 |
| author_facet | Crandall, Richard E. 1947-2012 Pomerance, Carl 1944- |
| author_role | aut aut |
| author_sort | Crandall, Richard E. 1947-2012 |
| author_variant | r e c re rec c p cp |
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| callnumber-first | Q - Science |
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| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 180 |
| classification_tum | MAT 100f |
| ctrlnum | (OCoLC)61318559 (DE-599)BVBBV023380567 |
| dewey-full | 512.723 512.7/23 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 512 - Algebra |
| dewey-raw | 512.723 512.7/23 |
| dewey-search | 512.723 512.7/23 |
| dewey-sort | 3512.723 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| discipline_str_mv | Mathematik |
| edition | 2. ed. |
| format | Book |
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| genre_facet | Lehrbuch |
| id | DE-604.BV023380567 |
| illustrated | Not Illustrated |
| index_date | 2024-07-02T21:16:14Z |
| indexdate | 2024-07-09T21:17:18Z |
| institution | BVB |
| isbn | 9780387252827 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016563679 |
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| physical | XV, 597 S. |
| publishDate | 2005 |
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| spelling | Crandall, Richard E. 1947-2012 Verfasser (DE-588)11390732X aut Prime numbers a computational perspective Richard Crandall ; Carl Pomerance 2. ed. New York [u.a.] Springer 2005 XV, 597 S. txt rdacontent n rdamedia nc rdacarrier Auch als Internetausgabe Getallen gtt Números primos larpcal Teoria dos números larpcal Numbers, Prime Informatik (DE-588)4026894-9 gnd rswk-swf Numerisches Verfahren (DE-588)4128130-5 gnd rswk-swf Primzahl (DE-588)4047263-2 gnd rswk-swf (DE-588)4123623-3 Lehrbuch gnd-content Primzahl (DE-588)4047263-2 s Numerisches Verfahren (DE-588)4128130-5 s DE-188 Informatik (DE-588)4026894-9 s 1\p DE-604 Pomerance, Carl 1944- Verfasser (DE-588)122920066 aut Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016563679&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Crandall, Richard E. 1947-2012 Pomerance, Carl 1944- Prime numbers a computational perspective Getallen gtt Números primos larpcal Teoria dos números larpcal Numbers, Prime Informatik (DE-588)4026894-9 gnd Numerisches Verfahren (DE-588)4128130-5 gnd Primzahl (DE-588)4047263-2 gnd |
| subject_GND | (DE-588)4026894-9 (DE-588)4128130-5 (DE-588)4047263-2 (DE-588)4123623-3 |
| title | Prime numbers a computational perspective |
| title_auth | Prime numbers a computational perspective |
| title_exact_search | Prime numbers a computational perspective |
| title_exact_search_txtP | Prime numbers a computational perspective |
| title_full | Prime numbers a computational perspective Richard Crandall ; Carl Pomerance |
| title_fullStr | Prime numbers a computational perspective Richard Crandall ; Carl Pomerance |
| title_full_unstemmed | Prime numbers a computational perspective Richard Crandall ; Carl Pomerance |
| title_short | Prime numbers |
| title_sort | prime numbers a computational perspective |
| title_sub | a computational perspective |
| topic | Getallen gtt Números primos larpcal Teoria dos números larpcal Numbers, Prime Informatik (DE-588)4026894-9 gnd Numerisches Verfahren (DE-588)4128130-5 gnd Primzahl (DE-588)4047263-2 gnd |
| topic_facet | Getallen Números primos Teoria dos números Numbers, Prime Informatik Numerisches Verfahren Primzahl Lehrbuch |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016563679&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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