From Fermat to Gauss: indefinite descent and methods of reduction in number theory:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Augsburg
Rauner
2006
|
| Series: | Algorismus
55 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | VII, 574 S. 21 cm |
| ISBN: | 3936905185 |
Staff View
MARC
| LEADER | 00000nam a2200000 cb4500 | ||
|---|---|---|---|
| 001 | BV021594511 | ||
| 003 | DE-604 | ||
| 005 | 20080527 | ||
| 007 | t | ||
| 008 | 060524s2006 |||| 00||| eng d | ||
| 020 | |a 3936905185 |9 3-936905-18-5 | ||
| 035 | |a (OCoLC)181560149 | ||
| 035 | |a (DE-599)BVBBV021594511 | ||
| 040 | |a DE-604 |b ger |e rakddb | ||
| 041 | 0 | |a eng | |
| 049 | |a DE-12 |a DE-210 |a DE-20 |a DE-37 |a DE-824 |a DE-634 |a DE-M100 |a DE-83 | ||
| 084 | |a SK 180 |0 (DE-625)143222: |2 rvk | ||
| 100 | 1 | |a Bussotti, Paolo |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a From Fermat to Gauss: indefinite descent and methods of reduction in number theory |c by Paolo Bussotti |
| 264 | 1 | |a Augsburg |b Rauner |c 2006 | |
| 300 | |a VII, 574 S. |c 21 cm | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Algorismus |v 55 | |
| 648 | 7 | |a Geschichte 1600-1900 |2 gnd |9 rswk-swf | |
| 650 | 4 | |a Geschichte | |
| 650 | 4 | |a Number theory | |
| 650 | 4 | |a Number theory |x History | |
| 650 | 0 | 7 | |a Deszendenzmethode |0 (DE-588)7534456-7 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Deszendenzmethode |0 (DE-588)7534456-7 |D s |
| 689 | 0 | 1 | |a Geschichte 1600-1900 |A z |
| 689 | 0 | |5 DE-604 | |
| 830 | 0 | |a Algorismus |v 55 |w (DE-604)BV001846336 |9 55 | |
| 856 | 4 | 2 | |m HEBIS Datenaustausch |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014809932&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
| 999 | |a oai:aleph.bib-bvb.de:BVB01-014809932 | ||
| 942 | 1 | 1 | |c 509 |e 22/bsb |f 0904 |
| 942 | 1 | 1 | |c 509 |e 22/bsb |f 0903 |
Record in the Search Index
| _version_ | 1804135372966330368 |
|---|---|
| adam_text | FROM FERMAT TO GAUSS: INDEFINITE DESCENT AND METHODS OF REDUCTION IN
NUMBER THEORY BY PAOLO BUSSOTTI DR. ERWIN RAUNER VERLAG AUGSBURG 2006
INDEX FOREWORD I PREFACE BY JOACHIM FISCHER III-VII CHAPTER 1:
INTRODUCTION, 1-16 1. WHAT IS THE INFINITE OR INDEFINITE DESCENT 1-4 2.
THE SET OF PROBLEMS DEALT WITH APPLYING THE DESCENT 4-12 3. INDEFINITE
DESCENT AND COMPLETE INDUCTION 12-13 4. AIMS AND STRUCTURE OF THE
PRESENT WORK 13-15 5. THE FIRST DOCUMENTED APPLICATION OF THE DESCENT 16
CHAPTER 2: FERMAT 17-1 84 1. RESEARCHES CONCERNING INTEGER NUMBERS IN
FERMAT S TIME 17-31 1.1. THE SCENE BEFORE FERMAT 17-20 1.2. THE
CONTRIBUTION OF FERMAT TO CLASSICAL PROBLEMS AND HIS NEW CONCEPTION
21-31 2. FERMAT AND THE INDEFINITE DESCENT 31-176 2.1. THE SOURCES 31-32
2.2. CASES IN WHICH FERMAT EXPLICITLY CLAIMED THAT HE HAD APPLIED HIS
OWN METHOD 32-39 2.3. THERE IS NO PYTHAGOREAN TRIANGLE OF WHICH THE
AREA IS EQUAL TO THE SQUARE OF AN INTEGER NUMBER 39-46 2.4. THE
APPLICATION OF THE INDEFINITE DESCENT TO THE AFFIRMATIVE THESES AND TO
THE PARTICULAR NEGATIVE THESES 46-175 2.4.1. EXPOSITION OF THE MATTER
46-48 2.4.2. EVERY PRIME NUMBER OF THE FORM 4N+L IS THE SUM OF TWO
SQUARES 49-77 2.4.2.1. THE DEMONSTRATION 49-57 2.4.2.2. LOGICAL
CONSIDERATIONS 58-65 2.4.2.3. POSSIBLE ATTRIBUTION OF PAOLINI S
DEMONSTRATION TO FERMAT 65-68 2.4.2.4. THE CONTINUED FRACTIONS IN
FERMAT S TIME 68-77 2.4.3. PELL S EQUATION 77-109 2.4.3.1. PREAMBLE
77-78 2.4.3.2. INTRODUCTION 79-81 2.4.3.3. FERMAT S DEFI AND THE
SOLUTION OF WALLIS AND BROUNCKER 81-97 2.4.3.4. SOME REMARKS 97-98
2.4.3.5. OTHER SOLUTIONS OF PELL S EQUATION 98-101 2.4.3.6. FERMAT S
REACTION TO THE SOLUTION BY WALLIS AND BROUNCKER 101-103 2.4.3.7. THE
GENERALIZATION OF PELL S EQUATION 103-109 2.4.4. THE THEOREM CONCERNING
THE POLYGONAL NUMBERS 109-171 2.4.4.1. THE PROTOHISTORY OF THE THEOREM
AND THE ASSERTIONS BY FERMAT 109-114 2.4.4.2. THE THREE TRIANGULARS
THEOREM 114-149 2.4.4.2.1. GENERAL CONSIDERATIONS 114-115 2.4.4.2.2.
NUMBERS OF THE FORM 8N+3 AND EXISTENCE OF THEIR MULTIPLES WHICH ARE THE
SUM OF THREE SQUARES 115-129 2.4.4.2.3. DEMONSTRATION THAT FOR EVERY
NUMBER OF THE FORM 8N+3 AN INFINITE NUMBER OF SQUARES A EXISTS SUCHTHAT
A 2 (8N + 3) = X 2 + Y 2 + Z 2 129-142 2.4.4.2.4. DEMONSTRATION THAT FOR
EVERY NUMBER OF THE FORM 8N+3 A SQUARE 5 R EXISTS SUCH THAT 5 2R (8N +
3) = X 2 +Y 2 +Z 2 142-146 2.4.4.2.5 CONCLUSIVE CONSIDERATIONS ON THE
THREE TRIANGULARS THEOREM 146-149 2.4.4.3. THE FOUR SQUARES THEOREM
149-171 2.4.4.3.1 PREAMBLE 149 2.4.4.3.2. THE DEMONSTRATION OF THE FOUR
SQUARES THEOREM 150-164 2.4.4.3.3. SOME REMARKS CONCERNING THE PREVIOUS
DEMONSTRATION 164-171 2.4.5. THE CUBIC CASE OF FERMAT S GREAT THEOREM
171-176 3. CONCLUSIONS 175-184 3A. PREAMBLE 176-177 3.2 THE
INTERCONNECTIONS BETWEEN FERMAT S THEOREMS 177-180 3.3. SOME OPINIONS
CONCERNING FERMAT 180-185 CHAPTER 3: FERMAT AND HIS SUCCESSORS : THE
MISFORTUNE BEHIND FERMAT S IDEA OF METHOD AND THE RED LINE
FERMAT-GAUSS-KRONECKER 187-213 1. PREAMBLE 187 2. THE MISFORTUNE BEHIND
FERMAT S IDEA OF METHOD 188-196 3. THE RELATIONS BETWEEN FERMAT AND THE
FOUNDATIONS OF MATHEMATICS 196-213 CHAPTER 4: EULER 215-292 1.
INTRODUCTION 215-217 2. BINARY QUADRATIC FORMS 220-260 2.1. THE THEOREM
CONCERNING THE PRIME NUMBERS OF THE FORM 4N+L 220-238 2.1.1. THE FIRST
VERSION OF THE PROOF 220-231 2.1.2. THE SECOND VERSION OF THE PROOF 231
-236 2.1.3. THE QUADRATIC CHARACTER OF -1 236-238 2.2. THE PRIME NUMBERS
OF THE FORM 8N+L AND 8N+3 AS SUMS OF SQUARES 238-244 2.2.1 THE
DIVISIBILITY OF THE FORM A 1 +2B 2 AND CONNECTION TO THE PRIME NUMBERS
OF THE FORM 8N+L AND 8N+3 238-244 2.3. PRIME NUMBERS OFTHE FORM 6N+L AS
SUMS OF SQUARES 245-250 2.3.1. INTRODUCTION 245-246 2.3.2. THE RESULTS
OBTAINED IN THE SUPPLEMENTUM 246-249 2.3.3. OTHER RESULTS OBTAINED BY
EULER FOR THE FORM X 2 +3Y 2 249-250 2.4. CONCLUSION CONCERNING THE
BINARY QUADRATIC FORMS 250-26 0 3. THE THEOREM CONCERNING POLYGONAL
NUMBERS 260-278 3.1. PREAMBLE 260-261 3.2. THE FOUR SQUARES THEOREM
261-273 3.2.1. EVERY NUMBER IS THE SUM OF FOUR INTEGER OR RATIONAL
SQUARES 262-266 3.2.2. EVERY INTEGER IS THE SUM OF FOUR INTEGER SQUARES
266-273 3.3. THE GENERAL THEOREM CONCERNING THE POLYGONAL NUMBERS AND
THE PARTITIO NUMERORUM 273-278 4. THE CUBIC CASE OF FERMAT S GREAT
THEOREM 278-287 4.1 PREAMBLE 278-279 4.2. EULER S DEMONSTRATION OF THE
IMPOSSIBILITY TO SOLVE IN INTEGERS THE EQUATION X 3 + Y 3 = Z 3 279-285
4.3. SOME REMARKS 285-287 5. CONCLUSIONS 287-292 CHAPTER 5: LAGRANGE 293
-417 1. INTRODUCTION 293-298 2. SOLUTION OF ALL UNDETERMINED EQUATIONS
OF SECOND DEGREE WITH TWO UNKNOWNS 299-341 2.1 THE STRUCTURE O/SUR LA
SOLUTION DES PROBLEMS INDETERMINES DU SECOND DEGRE 299 2.2. THE GENERAL
PROBLEM OF THE EQUATIONS OF SECOND DEGREE AND THE RATIONAL SOLUTIONS OF
THE EQUATION U 2 - BT 2 - A 299-311 2.3. THE INTEGER SOLUTIONS OF THE
EQUATION A = U 2 * BT 2 311-341 2.3.1. GENERAL CONSIDERATIONS 311-316
2.3.2. THE INTEGER SOLUTIONS OF THE EQUATION A = U - BT BEING B 0 AND
DIFFERENT FROM A SQUARE 316-335 2.3.3. FURTHER SPECIFICATIONS REGARDING
THE ROLE OF THE CONTINUED FRACTIONS 335-338 2.3.4. CONCLUSIVE
CONSIDERATIONS CONCERNING THE PAPER SUR LA SOLUTION DES PROBLEMES
INDETERMINES DU SECOND DEGRE 338-340 2.3.5. BRIEF NOTE ON THE CONCEPT OF
SOLUTION OF AN UNDETERMINED EQUATION 340-341 3. THE ADDITIONS TO EULER S
ALGEBRA 341-362 3.1. THE NEW METHOD METHOD IN ORDER TO SOLVE IN INTEGERS
ALL THE UNDETERMINED EQUATIONS OF SECOND DEGREE 343-362 3.1.1.
EXPOSITION OF THE METHOD 343-349 3.1.2. SOLUTION OF THE EQUATIONS OF
SECOND DEGREE AND BINARY QUADRATIC FORMS 349-352 3.1.3 FURTHER
SPECIFICATIONS CONCERNING THE EQUATION CY 2 * 2NYZ + BZ 2 = 1: BINARY
QUADRATIC FORMS, PELL S EQUATION AND PRINCIPLE OF THE SMALLEST INTEGER
352-362 4. THE RECHERCHES D ARITHMETIQUE 362-396 4.1. DISCUSSION OF THE
WORK 362-371 4.1.1. SUMMARY OF THE RESULTS 362-369 4.1.2. COMMENTARY ON
LAGRANGE S RESULTS AND CONCISE EXPOSITION OF THE METHODS USED IN THE
RECHERCHES 369-371 4.2. REDUCED FORMS AND THEIR EQUIVALENCE 372-396
4.2.1. NUMBER OF CLASSES IN WHICH THE REDUCED FORMS PY + IQYZ + RZ OF
DETERMINANT A * PR * Q 2 ARE DISTRIBUTED 372-381 4.2.2. THE PROBLEM OF
THE EQUIVALENCE BETWEEN FORMS HAVING POSITIVE DETERMINANT 382-396
4.2.2.1. THE DEMONSTRATION 382-390 4.2.2.2. REMARKS CONCERNING THE
PREVIOUS DEMONSTRATION 390-396 4.3. BRIEF CONCLUDING CONSIDERATIONS 396
5. THE FOUR SQUARES THEOREM 396-404 6 THE EQUATION X 4 -2Y 4 =Z 2
405-412 7. CONCLUSIONS 412-417 7.1. CONCLUDING CONSIDERATIONS CONCERNING
LAGRANGE S METHODS 412-416 7.2. LAGRANGE AND THE RIGOURISM 416-417
CHAPTER 6: GAUSS 419-439 1. INTRODUCTION 419-423 2. BINARY REDUCED
QUADRATIC FORMS AND INDEFINITE DESCENT 424-427 3. GAUSS PROOF OF
FERMAT S GREAT THEOREM FOR EXPONENTS 3 AND 5 427-437 3.1. THE
DEMONSTRATION OF THE CUBIC CASE 427-433 3.2 SOME REMARKS CONCERNING
GAUSS DEMONSTRATION 434-437 4. CONCLUSIONS 437-439 CHAPTER 7: LOGICAL
CONSIDERATIONS 441-472 1. INTRODUCTION 441-443 2. INDEFINITE DESCENT AND
AFFIRMATIVE THESES 443-447 3. CLASSIFICATION OF THE ANALYSED PROOFS
447-449 4. CONSIDERATIONS CONCERNING SOMEONE OF THE PROPOSED
DEMONSTRATIONS 450-469 4.1. THE THEOREMS OF NUMBER 6 450-451 4.2. THE
PROPOSITIONS DEMONSTRATED BY REDUCTION-DESCENT AND THE PRINCIPLE OF THE
SMALLEST NUMBER 451-454 4.3 INDEFINITE DESCENT AND PRINCIPLE OF THE
SMALLEST NUMBER 454-456 4.4. INDEFINITE DESCENT AND COMPLETE INDUCTION
456-469 5. CONCLUSIONS 469-472 CHAPTER 8: CONCLUSION 473-479 APPENDIX BY
SERGIO PAOLINI 481-554 PREAMBLE 481-482 1. EVERY PRIME NUMBER OF THE
FORM 4N+L IS THE SUM OF TWO SQUARES. FIRST DEMONSTRATION 482-495 2. THE
DECOMPOSITION OF A PRIME NUMBER IN SUM OF TWO SQUARES 495-507 A) PRIME
NUMBERS OF THE FORM 4N+L 495-499 B) PRIME NUMBERS OF THE FORM 8N+3
499-504 C) PRIME NUMBERS OF THE FORM 3N+L 504-507 3. THE THREE
TRIANGULARS THEOREM 507-534 4. THE POLYGONAL NUMBERS 534-547 5. NO
INTEGER CUBE IS THE SUM OF TWO CUBES 547-554 BIBLIOGRAPHY 555-569 INDEX
OF NAMES 571 -574
|
| adam_txt |
FROM FERMAT TO GAUSS: INDEFINITE DESCENT AND METHODS OF REDUCTION IN
NUMBER THEORY BY PAOLO BUSSOTTI DR. ERWIN RAUNER VERLAG AUGSBURG 2006
INDEX FOREWORD I PREFACE BY JOACHIM FISCHER III-VII CHAPTER 1:
INTRODUCTION, 1-16 1. WHAT IS THE INFINITE OR INDEFINITE DESCENT 1-4 2.
THE SET OF PROBLEMS DEALT WITH APPLYING THE DESCENT 4-12 3. INDEFINITE
DESCENT AND COMPLETE INDUCTION 12-13 4. AIMS AND STRUCTURE OF THE
PRESENT WORK 13-15 5. THE FIRST DOCUMENTED APPLICATION OF THE DESCENT 16
CHAPTER 2: FERMAT 17-1 84 1. RESEARCHES CONCERNING INTEGER NUMBERS IN
FERMAT'S TIME 17-31 1.1. THE SCENE BEFORE FERMAT 17-20 1.2. THE
CONTRIBUTION OF FERMAT TO "CLASSICAL" PROBLEMS AND HIS NEW CONCEPTION
21-31 2. FERMAT AND THE INDEFINITE DESCENT 31-176 2.1. THE SOURCES 31-32
2.2. CASES IN WHICH FERMAT EXPLICITLY CLAIMED THAT HE HAD APPLIED HIS
OWN METHOD 32-39 2.3. "THERE IS NO PYTHAGOREAN TRIANGLE OF WHICH THE
AREA IS EQUAL TO THE SQUARE OF AN INTEGER NUMBER" 39-46 2.4. THE
APPLICATION OF THE INDEFINITE DESCENT TO THE AFFIRMATIVE THESES AND TO
THE "PARTICULAR" NEGATIVE THESES 46-175 2.4.1. EXPOSITION OF THE MATTER
46-48 2.4.2. EVERY PRIME NUMBER OF THE FORM 4N+L IS THE SUM OF TWO
SQUARES 49-77 2.4.2.1. THE DEMONSTRATION 49-57 2.4.2.2. LOGICAL
CONSIDERATIONS 58-65 2.4.2.3. POSSIBLE ATTRIBUTION OF PAOLINI'S
DEMONSTRATION TO FERMAT 65-68 2.4.2.4. THE CONTINUED FRACTIONS IN
FERMAT'S TIME 68-77 2.4.3. PELL'S EQUATION 77-109 2.4.3.1. PREAMBLE
77-78 2.4.3.2. INTRODUCTION 79-81 2.4.3.3. FERMAT'S "DEFI" AND THE
SOLUTION OF WALLIS AND BROUNCKER 81-97 2.4.3.4. SOME REMARKS 97-98
2.4.3.5. OTHER SOLUTIONS OF PELL'S EQUATION 98-101 2.4.3.6. FERMAT'S
REACTION TO THE SOLUTION BY WALLIS AND BROUNCKER 101-103 2.4.3.7. THE
GENERALIZATION OF PELL'S EQUATION 103-109 2.4.4. THE THEOREM CONCERNING
THE POLYGONAL NUMBERS 109-171 2.4.4.1. THE "PROTOHISTORY" OF THE THEOREM
AND THE ASSERTIONS BY FERMAT 109-114 2.4.4.2. THE THREE TRIANGULARS
THEOREM 114-149 2.4.4.2.1. GENERAL CONSIDERATIONS 114-115 2.4.4.2.2.
NUMBERS OF THE FORM 8N+3 AND EXISTENCE OF THEIR MULTIPLES WHICH ARE THE
SUM OF THREE SQUARES 115-129 2.4.4.2.3. DEMONSTRATION THAT FOR EVERY
NUMBER OF THE FORM 8N+3 AN INFINITE NUMBER OF SQUARES A EXISTS SUCHTHAT
A 2 (8N + 3) = X 2 + Y 2 + Z 2 129-142 2.4.4.2.4. DEMONSTRATION THAT FOR
EVERY NUMBER OF THE FORM 8N+3 A SQUARE 5 R EXISTS SUCH THAT 5 2R (8N +
3) = X 2 +Y 2 +Z 2 142-146 2.4.4.2.5 CONCLUSIVE CONSIDERATIONS ON THE
THREE TRIANGULARS THEOREM 146-149 2.4.4.3. THE FOUR SQUARES THEOREM
149-171 2.4.4.3.1 PREAMBLE 149 2.4.4.3.2. THE DEMONSTRATION OF THE FOUR
SQUARES THEOREM 150-164 2.4.4.3.3. SOME REMARKS CONCERNING THE PREVIOUS
DEMONSTRATION 164-171 2.4.5. THE CUBIC CASE OF "FERMAT'S GREAT THEOREM"
171-176 3. CONCLUSIONS 175-184 3A. PREAMBLE 176-177 3.2 THE
INTERCONNECTIONS BETWEEN FERMAT'S THEOREMS 177-180 3.3. SOME OPINIONS
CONCERNING FERMAT 180-185 CHAPTER 3: FERMAT AND HIS "SUCCESSORS": THE
MISFORTUNE BEHIND FERMAT'S IDEA OF METHOD AND THE "RED LINE"
FERMAT-GAUSS-KRONECKER 187-213 1. PREAMBLE 187 2. THE MISFORTUNE BEHIND
FERMAT'S IDEA OF METHOD 188-196 3. THE RELATIONS BETWEEN FERMAT AND THE
FOUNDATIONS OF MATHEMATICS 196-213 CHAPTER 4: EULER 215-292 1.
INTRODUCTION 215-217 2. BINARY QUADRATIC FORMS 220-260 2.1. THE THEOREM
CONCERNING THE PRIME NUMBERS OF THE FORM 4N+L 220-238 2.1.1. THE FIRST
VERSION OF THE PROOF 220-231 2.1.2. THE SECOND VERSION OF THE PROOF 231
-236 2.1.3. THE QUADRATIC CHARACTER OF -1 236-238 2.2. THE PRIME NUMBERS
OF THE FORM 8N+L AND 8N+3 AS SUMS OF SQUARES 238-244 2.2.1 THE
DIVISIBILITY OF THE FORM A 1 +2B 2 AND CONNECTION TO THE PRIME NUMBERS
OF THE FORM 8N+L AND 8N+3 238-244 2.3. PRIME NUMBERS OFTHE FORM 6N+L AS
SUMS OF SQUARES 245-250 2.3.1. INTRODUCTION 245-246 2.3.2. THE RESULTS
OBTAINED IN THE SUPPLEMENTUM 246-249 2.3.3. OTHER RESULTS OBTAINED BY
EULER FOR THE FORM X 2 +3Y 2 249-250 2.4. CONCLUSION CONCERNING THE
BINARY QUADRATIC FORMS 250-26 0 3. THE THEOREM CONCERNING POLYGONAL
NUMBERS 260-278 3.1. PREAMBLE 260-261 3.2. THE FOUR SQUARES THEOREM
261-273 3.2.1. EVERY NUMBER IS THE SUM OF FOUR INTEGER OR RATIONAL
SQUARES 262-266 3.2.2. EVERY INTEGER IS THE SUM OF FOUR INTEGER SQUARES
266-273 3.3. THE GENERAL THEOREM CONCERNING THE POLYGONAL NUMBERS AND
THE "PARTITIO NUMERORUM" 273-278 4. THE CUBIC CASE OF "FERMAT'S GREAT
THEOREM" 278-287 4.1 PREAMBLE 278-279 4.2. EULER'S DEMONSTRATION OF THE
IMPOSSIBILITY TO SOLVE IN INTEGERS THE EQUATION X 3 + Y 3 = Z 3 279-285
4.3. SOME REMARKS 285-287 5. CONCLUSIONS 287-292 CHAPTER 5: LAGRANGE 293
-417 1. INTRODUCTION 293-298 2. SOLUTION OF ALL UNDETERMINED EQUATIONS
OF SECOND DEGREE WITH TWO UNKNOWNS 299-341 2.1 THE STRUCTURE O/SUR LA
SOLUTION DES PROBLEMS INDETERMINES DU SECOND DEGRE 299 2.2. THE GENERAL
PROBLEM OF THE EQUATIONS OF SECOND DEGREE AND THE RATIONAL SOLUTIONS OF
THE EQUATION U 2 - BT 2 - A 299-311 2.3. THE INTEGER SOLUTIONS OF THE
EQUATION A = U 2 * BT 2 311-341 2.3.1. GENERAL CONSIDERATIONS 311-316
2.3.2. THE INTEGER SOLUTIONS OF THE EQUATION A = U - BT BEING B 0 AND
DIFFERENT FROM A SQUARE 316-335 2.3.3. FURTHER SPECIFICATIONS REGARDING
THE ROLE OF THE CONTINUED FRACTIONS 335-338 2.3.4. CONCLUSIVE
CONSIDERATIONS CONCERNING THE PAPER SUR LA SOLUTION DES PROBLEMES
INDETERMINES DU SECOND DEGRE 338-340 2.3.5. BRIEF NOTE ON THE CONCEPT OF
SOLUTION OF AN UNDETERMINED EQUATION 340-341 3. THE ADDITIONS TO EULER'S
ALGEBRA 341-362 3.1. THE NEW METHOD METHOD IN ORDER TO SOLVE IN INTEGERS
ALL THE UNDETERMINED EQUATIONS OF SECOND DEGREE 343-362 3.1.1.
EXPOSITION OF THE METHOD 343-349 3.1.2. SOLUTION OF THE EQUATIONS OF
SECOND DEGREE AND BINARY QUADRATIC FORMS 349-352 3.1.3 FURTHER
SPECIFICATIONS CONCERNING THE EQUATION CY 2 * 2NYZ + BZ 2 = 1: BINARY
QUADRATIC FORMS, PELL'S EQUATION AND PRINCIPLE OF THE SMALLEST INTEGER
352-362 4. THE RECHERCHES D'ARITHMETIQUE 362-396 4.1. DISCUSSION OF THE
WORK 362-371 4.1.1. SUMMARY OF THE RESULTS 362-369 4.1.2. COMMENTARY ON
LAGRANGE 'S RESULTS AND CONCISE EXPOSITION OF THE METHODS USED IN THE
RECHERCHES 369-371 4.2. REDUCED FORMS AND THEIR EQUIVALENCE 372-396
4.2.1. NUMBER OF CLASSES IN WHICH THE REDUCED FORMS PY + IQYZ + RZ OF
DETERMINANT A * PR * Q 2 ARE DISTRIBUTED 372-381 4.2.2. THE PROBLEM OF
THE EQUIVALENCE BETWEEN FORMS HAVING POSITIVE DETERMINANT 382-396
4.2.2.1. THE DEMONSTRATION 382-390 4.2.2.2. REMARKS CONCERNING THE
PREVIOUS DEMONSTRATION 390-396 4.3. BRIEF CONCLUDING CONSIDERATIONS 396
5. THE FOUR SQUARES THEOREM 396-404 6 THE EQUATION X 4 -2Y 4 =Z 2
405-412 7. CONCLUSIONS 412-417 7.1. CONCLUDING CONSIDERATIONS CONCERNING
LAGRANGE 'S METHODS 412-416 7.2. LAGRANGE AND THE RIGOURISM 416-417
CHAPTER 6: GAUSS 419-439 1. INTRODUCTION 419-423 2. BINARY REDUCED
QUADRATIC FORMS AND INDEFINITE DESCENT 424-427 3. GAUSS' PROOF OF
"FERMAT'S GREAT THEOREM " FOR EXPONENTS 3 AND 5 427-437 3.1. THE
DEMONSTRATION OF THE CUBIC CASE 427-433 3.2 SOME REMARKS CONCERNING
GAUSS' DEMONSTRATION 434-437 4. CONCLUSIONS 437-439 CHAPTER 7: LOGICAL
CONSIDERATIONS 441-472 1. INTRODUCTION 441-443 2. INDEFINITE DESCENT AND
AFFIRMATIVE THESES 443-447 3. CLASSIFICATION OF THE ANALYSED PROOFS
447-449 4. CONSIDERATIONS CONCERNING SOMEONE OF THE PROPOSED
DEMONSTRATIONS 450-469 4.1. THE THEOREMS OF NUMBER 6 450-451 4.2. THE
PROPOSITIONS DEMONSTRATED BY REDUCTION-DESCENT AND THE PRINCIPLE OF THE
SMALLEST NUMBER 451-454 4.3 INDEFINITE DESCENT AND PRINCIPLE OF THE
SMALLEST NUMBER 454-456 4.4. INDEFINITE DESCENT AND COMPLETE INDUCTION
456-469 5. CONCLUSIONS 469-472 CHAPTER 8: CONCLUSION 473-479 APPENDIX BY
SERGIO PAOLINI 481-554 PREAMBLE 481-482 1. EVERY PRIME NUMBER OF THE
FORM 4N+L IS THE SUM OF TWO SQUARES. FIRST DEMONSTRATION 482-495 2. THE
DECOMPOSITION OF A PRIME NUMBER IN SUM OF TWO SQUARES 495-507 A) PRIME
NUMBERS OF THE FORM 4N+L 495-499 B) PRIME NUMBERS OF THE FORM 8N+3
499-504 C) PRIME NUMBERS OF THE FORM 3N+L 504-507 3. THE THREE
TRIANGULARS THEOREM 507-534 4. THE POLYGONAL NUMBERS 534-547 5. NO
INTEGER CUBE IS THE SUM OF TWO CUBES 547-554 BIBLIOGRAPHY 555-569 INDEX
OF NAMES 571 -574 |
| any_adam_object | 1 |
| any_adam_object_boolean | 1 |
| author | Bussotti, Paolo |
| author_facet | Bussotti, Paolo |
| author_role | aut |
| author_sort | Bussotti, Paolo |
| author_variant | p b pb |
| building | Verbundindex |
| bvnumber | BV021594511 |
| classification_rvk | SK 180 |
| ctrlnum | (OCoLC)181560149 (DE-599)BVBBV021594511 |
| discipline | Mathematik |
| discipline_str_mv | Mathematik |
| era | Geschichte 1600-1900 gnd |
| era_facet | Geschichte 1600-1900 |
| format | Book |
| fullrecord | <?xml version="1.0" encoding="UTF-8"?><collection xmlns="http://www.loc.gov/MARC21/slim"><record><leader>01577nam a2200421 cb4500</leader><controlfield tag="001">BV021594511</controlfield><controlfield tag="003">DE-604</controlfield><controlfield tag="005">20080527 </controlfield><controlfield tag="007">t</controlfield><controlfield tag="008">060524s2006 |||| 00||| eng d</controlfield><datafield tag="020" ind1=" " ind2=" "><subfield code="a">3936905185</subfield><subfield code="9">3-936905-18-5</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(OCoLC)181560149</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(DE-599)BVBBV021594511</subfield></datafield><datafield tag="040" ind1=" " ind2=" "><subfield code="a">DE-604</subfield><subfield code="b">ger</subfield><subfield code="e">rakddb</subfield></datafield><datafield tag="041" ind1="0" ind2=" "><subfield code="a">eng</subfield></datafield><datafield tag="049" ind1=" " ind2=" "><subfield code="a">DE-12</subfield><subfield code="a">DE-210</subfield><subfield code="a">DE-20</subfield><subfield code="a">DE-37</subfield><subfield code="a">DE-824</subfield><subfield code="a">DE-634</subfield><subfield code="a">DE-M100</subfield><subfield code="a">DE-83</subfield></datafield><datafield tag="084" ind1=" " ind2=" "><subfield code="a">SK 180</subfield><subfield code="0">(DE-625)143222:</subfield><subfield code="2">rvk</subfield></datafield><datafield tag="100" ind1="1" ind2=" "><subfield code="a">Bussotti, Paolo</subfield><subfield code="e">Verfasser</subfield><subfield code="4">aut</subfield></datafield><datafield tag="245" ind1="1" ind2="0"><subfield code="a">From Fermat to Gauss: indefinite descent and methods of reduction in number theory</subfield><subfield code="c">by Paolo Bussotti</subfield></datafield><datafield tag="264" ind1=" " ind2="1"><subfield code="a">Augsburg</subfield><subfield code="b">Rauner</subfield><subfield code="c">2006</subfield></datafield><datafield tag="300" ind1=" " ind2=" "><subfield code="a">VII, 574 S.</subfield><subfield code="c">21 cm</subfield></datafield><datafield tag="336" ind1=" " ind2=" "><subfield code="b">txt</subfield><subfield code="2">rdacontent</subfield></datafield><datafield tag="337" ind1=" " ind2=" "><subfield code="b">n</subfield><subfield code="2">rdamedia</subfield></datafield><datafield tag="338" ind1=" " ind2=" "><subfield code="b">nc</subfield><subfield code="2">rdacarrier</subfield></datafield><datafield tag="490" ind1="1" ind2=" "><subfield code="a">Algorismus</subfield><subfield code="v">55</subfield></datafield><datafield tag="648" ind1=" " ind2="7"><subfield code="a">Geschichte 1600-1900</subfield><subfield code="2">gnd</subfield><subfield code="9">rswk-swf</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Geschichte</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Number theory</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Number theory</subfield><subfield code="x">History</subfield></datafield><datafield tag="650" ind1="0" ind2="7"><subfield code="a">Deszendenzmethode</subfield><subfield code="0">(DE-588)7534456-7</subfield><subfield code="2">gnd</subfield><subfield code="9">rswk-swf</subfield></datafield><datafield tag="689" ind1="0" ind2="0"><subfield code="a">Deszendenzmethode</subfield><subfield code="0">(DE-588)7534456-7</subfield><subfield code="D">s</subfield></datafield><datafield tag="689" ind1="0" ind2="1"><subfield code="a">Geschichte 1600-1900</subfield><subfield code="A">z</subfield></datafield><datafield tag="689" ind1="0" ind2=" "><subfield code="5">DE-604</subfield></datafield><datafield tag="830" ind1=" " ind2="0"><subfield code="a">Algorismus</subfield><subfield code="v">55</subfield><subfield code="w">(DE-604)BV001846336</subfield><subfield code="9">55</subfield></datafield><datafield tag="856" ind1="4" ind2="2"><subfield code="m">HEBIS Datenaustausch</subfield><subfield code="q">application/pdf</subfield><subfield code="u">http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014809932&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA</subfield><subfield code="3">Inhaltsverzeichnis</subfield></datafield><datafield tag="999" ind1=" " ind2=" "><subfield code="a">oai:aleph.bib-bvb.de:BVB01-014809932</subfield></datafield><datafield tag="942" ind1="1" ind2="1"><subfield code="c">509</subfield><subfield code="e">22/bsb</subfield><subfield code="f">0904</subfield></datafield><datafield tag="942" ind1="1" ind2="1"><subfield code="c">509</subfield><subfield code="e">22/bsb</subfield><subfield code="f">0903</subfield></datafield></record></collection> |
| id | DE-604.BV021594511 |
| illustrated | Not Illustrated |
| index_date | 2024-07-02T14:45:33Z |
| indexdate | 2024-07-09T20:39:28Z |
| institution | BVB |
| isbn | 3936905185 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-014809932 |
| oclc_num | 181560149 |
| open_access_boolean | |
| owner | DE-12 DE-210 DE-20 DE-37 DE-824 DE-634 DE-M100 DE-83 |
| owner_facet | DE-12 DE-210 DE-20 DE-37 DE-824 DE-634 DE-M100 DE-83 |
| physical | VII, 574 S. 21 cm |
| publishDate | 2006 |
| publishDateSearch | 2006 |
| publishDateSort | 2006 |
| publisher | Rauner |
| record_format | marc |
| series | Algorismus |
| series2 | Algorismus |
| spelling | Bussotti, Paolo Verfasser aut From Fermat to Gauss: indefinite descent and methods of reduction in number theory by Paolo Bussotti Augsburg Rauner 2006 VII, 574 S. 21 cm txt rdacontent n rdamedia nc rdacarrier Algorismus 55 Geschichte 1600-1900 gnd rswk-swf Geschichte Number theory Number theory History Deszendenzmethode (DE-588)7534456-7 gnd rswk-swf Deszendenzmethode (DE-588)7534456-7 s Geschichte 1600-1900 z DE-604 Algorismus 55 (DE-604)BV001846336 55 HEBIS Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014809932&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Bussotti, Paolo From Fermat to Gauss: indefinite descent and methods of reduction in number theory Algorismus Geschichte Number theory Number theory History Deszendenzmethode (DE-588)7534456-7 gnd |
| subject_GND | (DE-588)7534456-7 |
| title | From Fermat to Gauss: indefinite descent and methods of reduction in number theory |
| title_auth | From Fermat to Gauss: indefinite descent and methods of reduction in number theory |
| title_exact_search | From Fermat to Gauss: indefinite descent and methods of reduction in number theory |
| title_exact_search_txtP | From Fermat to Gauss: indefinite descent and methods of reduction in number theory |
| title_full | From Fermat to Gauss: indefinite descent and methods of reduction in number theory by Paolo Bussotti |
| title_fullStr | From Fermat to Gauss: indefinite descent and methods of reduction in number theory by Paolo Bussotti |
| title_full_unstemmed | From Fermat to Gauss: indefinite descent and methods of reduction in number theory by Paolo Bussotti |
| title_short | From Fermat to Gauss: indefinite descent and methods of reduction in number theory |
| title_sort | from fermat to gauss indefinite descent and methods of reduction in number theory |
| topic | Geschichte Number theory Number theory History Deszendenzmethode (DE-588)7534456-7 gnd |
| topic_facet | Geschichte Number theory Number theory History Deszendenzmethode |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014809932&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV001846336 |
| work_keys_str_mv | AT bussottipaolo fromfermattogaussindefinitedescentandmethodsofreductioninnumbertheory |