ΠΡΩΤΟΙ ΑΡΙΘΜΟΙ ΚΑΙ ΚΥΚΛΟΤΟΜΙΑ-ΠΡΩΤΟΙ ΑΡΙΘΜΟΙ ΤΗΣ ΜΟΡΦΗΣ Χ + (Χ+1)

THE PRESENT DOCTORAL THESIS CONSISTS OF THREE CHAPTERS, NAMELY: CHAPTER 1: CRITERIA AND FORMULAE FOR PRIME NUMBERS. CHAPTER 2: PRIMES AND CYCLOTOMY. CHAPTER3: PRIME AND COMPOSITE NUMBERS OF THE FORM X2+(X+1)2. IN CHAPTER 1 NEW PRIMALITY CRITERIA ARE OBTAINED, ALONG WITH NEW FORMULAE FOR THE SEQUENCE OF PRIMES (RECURSIVE AND EXPLICIT). SUCH FORMULAE ARE EXTENDED IN CHAPTER 2 BY MEANS OF CYCLOTOMIC CONCEPTS AND COSINE FUNCTIONS. IN CHAPTER 3 THE PRIME AND COMPOSITENUMBERS OF THE FORM X2+(X+1)2 ARE EXAMINED. THEIR STUDY DEPENDS HEAVILY ON THE FOLLOWING THEOREM(SIERPINSKI): THE NUMBER X2+(X+1)2 IS COMPOSITE IF THERE EXIST NATURAL NUMBERS Y, Z SUCH THAT: (*) T(X)= T(Y)+T(Z). (HERE T(X), T(Y), T(Z) DENOTE TRIANGULAR NUMBERS). THE DESCRIPTION OF ALL COMPOSITE NUMBERS OF THEFORM X2+(X+1)2 IS REDUCED TO THE STUDY OF THE INTEGRAL SOLUTIONS OF THE FOLLOWING FAMILY OF DIOPHANTINE EQUATIONS OF FERMAT TYPE: (FK) X2- 2Y2=2K2-1, K=0,1,2,... . THUS THE STUDY OF EQUATION (*) IS REDUCED TO THE STUDY OF THE FAMILY OF EQUATIONS (FK) IN TERMS OF GAUSS TYPE TRANSFORMATIONS. FINALLY, ALL COMPOSITE NUMBERS OF THE FORM X2+(X+1)2 ARE DETERMINED IN TERMS OF SUITABLE (NON-HOMOGENEOUS) LINEAR RECURRENCE SEQUENCES OF ORDER 2 (THEOREM 3.4.15). AS A CONSEQUENCE, ALL PRIMES OF THE SAME FORM IN A GIVEN INTERVAL CAN BE DETERMINED BY A SEIVING PROCEDURE (THEOREM 3.4.16).

Grantor dc:publisher

National and Kapodistrian University of Athens

Year dc:date

1986

Author dc:creator

• Τσαγκάρης, Παναγιώτης

• ΑΠΑΛΕΙΦΟΥΣΑ ΣΥΝΑΡΜΟΖΟΥΣΑ
• ΑΡΙΘΜΗΤΙΚΕΣ ΣΥΝΑΡΤΗΣΕΙΣ
• ΑΡΧΙΚΕΣ ΡΙΖΕΣ ΤΗΣ ΜΟΝΑΔΑΣ
• Διάσπαση
• Διοφαντικές εξισώσεις
• Κυκλοτομία
• ΜΑΘΗΜΑΤΙΚΑ {ΘΕΩΡΙΑ ΑΡΙΘΜΩΝ}
• ΣΥΝΔΕΣΜΙΚΑ ΣΗΜΕΙΑ
• ΤΡΙΓΩΝΟΙ ΑΡΙΘΜΟΙ
• ARITHMETICAL FUNCTIONS
• Cyclotomy
• DIOPHANTINE EQUATIONS
• LATTICE POINTS
• PRIMITIVE ROOTS OF UNITY
• Resultant
• TRIANGULAR NUMBERS-DECOMPOSITION
• Φυσικές Επιστήμες
• Μαθηματικά
• Natural Sciences
• Mathematics

Language dc:language

gre

Identifier

10.12681/eadd/0150

OAI identifier oai:identifier

oai:10442/0150