An-Najah University Journal for Research - A (Natural Sciences)

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Keywords

• Korselt numbers
• Korselt sets
• Carmichael numbers

Abstract

The Korselt numbers and sets were discussed for the first time in 2007. The problem can be considered as a new one with limited literature making it as a new field of research. Let N be a positive integer and α a non-zero integer. If N ≠ α and p divides N for each prime divisor p of N, then N is called an α−Korselt number (Kα-number). In this thesis, many concepts such as Korselt numbers that are related to Carmichael numbers have been studied. Korselt determined Korselt numbers by studying the converse of Fermat’s Little theorem and it can be noticed that all these numbers depend on number theory, prime numbers, divisibility and modular arithmetic. To validate the concerned theorems, an illustrated proofs were followed through detailed steps in addition to many examples are solved in order to support the correctness of these theories. It is important to say that some errors in literature were addressed by us. Consequently, we introduced proper corrections for them. Finally, many notes have been taken and directed us to build and develop a number of complicated algorithms, some of them in order to find Korselt sets for relatively large numbers in an effective way in a short time which may require a great time and need tedious effort if it is to be calculated manually

معلومات المقال

قيد النشر

الكلمات الإفتتاحية

• Korselt numbers
• Korselt sets
• Carmichael numbers

الملخص

The Korselt numbers and sets were discussed for the first time in 2007. The problem can be considered as a new one with limited literature making it as a new field of research. Let N be a positive integer and α a non-zero integer. If N ≠ α and p divides N for each prime divisor p of N, then N is called an α−Korselt number (Kα-number). In this thesis, many concepts such as Korselt numbers that are related to Carmichael numbers have been studied. Korselt determined Korselt numbers by studying the converse of Fermat’s Little theorem and it can be noticed that all these numbers depend on number theory, prime numbers, divisibility and modular arithmetic. To validate the concerned theorems, an illustrated proofs were followed through detailed steps in addition to many examples are solved in order to support the correctness of these theories. It is important to say that some errors in literature were addressed by us. Consequently, we introduced proper corrections for them. Finally, many notes have been taken and directed us to build and develop a number of complicated algorithms, some of them in order to find Korselt sets for relatively large numbers in an effective way in a short time which may require a great time and need tedious effort if it is to be calculated manually