EVEN PERFECT NUMBERS AND THEIR EULER ’ S FUNCTION

The purpose of this article is to prove some results on even perfect numbers and on their Euler’s function. The results obtained are all straightforward deductions from well-known elementary number theory.


INTRODUCTION.
A positive integer is called a perfect number if it is equal to the sum of its positive divisors excluding itself.
The n th triangular number is the sum of the first n-positive integers F .n k I n(n+l) T(n).k=l Euler's function @(n) is the number of positive integers less than or equal to n and relatively prime to n.
The number of divisors function d(n) is the number of positive divisors of MAIN RESULTS.
The proof of the following Theorem can be found in many elementary number theory books; see, for example, [l:p.98].
If n is an even perfect number, there exists a prime 2P-1 such that n 2p-1(2p-1).
THEOREM 2. If T(pl) is any even perfect number, where Pl is prime, and if Pk is the first prime in the sequence {P2, P3 Pj } where pj 2Pj_1+1, then T(Pk) is the next even perfect number.

PROOF.
It follows from Theorem 1 that an even perfect number is of the form 2n-l(2n-l), where 2n-I is prime.Now, 2n-l(2n-l) can be written as T(pI), where pl =2n-1.Let Pi be any composite term of the sequence ..... 2 n+i-1 It can be shown that Pi -1, using the facts Pi 2n-1' and 2Pj_1+1.Now, it follows from Theorem that T(Pi) 2n+i-2(2 n+i-1 -I) is Pj not an even perfect number.
If n is an even perfect number 2P-1(2P-1), then The proof of the followingTheorem 4 can also be found in many elementary number theory books; see, for example 1: p. 63.As a consequence of Theorem 4, one can easily obtain Theorem 5, Corollary 2, and Corollary 3 THEOREM 5. n 2P-1(2 p 1) is an even perfect number if and only if #(.) P-( 2-). ,h, P-pi.COROLLARY 2.
which i=l implies that (2 p -1) divides exactly one of the factors (l+ai), 1 i k, say (1 + aj).Thus (1 + aj) (2 p -1) X for some X and exactly one j such that k J k, and (2 p -1) {1 + a i) 2 p-1 (2 p -1), that is, i=l k U X {1 / t) 2p-l, which tmplie that / i 2 # J, I > O; X J gj 0 and i p-I which is i=1 k Observe that i k-1 since i > 0 for i J and O.

Thus, i=1
p-1 k-1 or p k, which is (t).Now, {1 + aj) {2 p -1) for exactly one , such that 1 J k and j 0 implies that aj 2 J (2 p 1) 1 for exactly one J such that 1 k and O, which proves a Finally, + a 2

Call for Papers
As a multidisciplinary field, financial engineering is becoming increasingly important in today's economic and financial world, especially in areas such as portfolio management, asset valuation and prediction, fraud detection, and credit risk management.For example, in a credit risk context, the recently approved Basel II guidelines advise financial institutions to build comprehensible credit risk models in order to optimize their capital allocation policy.Computational methods are being intensively studied and applied to improve the quality of the financial decisions that need to be made.Until now, computational methods and models are central to the analysis of economic and financial decisions.However, more and more researchers have found that the financial environment is not ruled by mathematical distributions or statistical models.In such situations, some attempts have also been made to develop financial engineering models using intelligent computing approaches.For example, an artificial neural network (ANN) is a nonparametric estimation technique which does not make any distributional assumptions regarding the underlying asset.Instead, ANN approach develops a model using sets of unknown parameters and lets the optimization routine seek the best fitting parameters to obtain the desired results.The main aim of this special issue is not to merely illustrate the superior performance of a new intelligent computational method, but also to demonstrate how it can be used effectively in a financial engineering environment to improve and facilitate financial decision making.In this sense, the submissions should especially address how the results of estimated computational models (e.g., ANN, support vector machines, evolutionary algorithm, and fuzzy models) can be used to develop intelligent, easy-to-use, and/or comprehensible computational systems (e.g., decision support systems, agent-based system, and web-based systems) This special issue will include (but not be limited to) the following topics: • Computational methods: artificial intelligence, neural networks, evolutionary algorithms, fuzzy inference, hybrid learning, ensemble learning, cooperative learning, multiagent learning

•
Application fields: asset valuation and prediction, asset allocation and portfolio selection, bankruptcy prediction, fraud detection, credit risk management • Implementation aspects: decision support systems, expert systems, information systems, intelligent agents, web service, monitoring, deployment, implementation