"One of the main nuclear features which led to the development of the shell structure is the existence of what are usually called the magic numbers.That such numbers exist was first remarked by Elsasser in 1933. What makes a number magic is that a configuration of a magic number of neutrons,or of protons, is unusually stable whatever the associated number of the other nucleons. When Teller and I worked on a paper on the origin of the elements, I stumbled over the magic number.We found that there were a few nuclei which have a greater isotopic as well as cosmic abundance than our theory of any other reasonable continum theory could possible explain.Then I found that those nuclei had something in common:they either had 82 neutrons, whatever the associated proton number, or 50 neutrons.Eighty-two and fiftu are magic numbers. That nuclei of this type are unusually abundant indicates that the excess stability must have played a part in the process of the creation of the elements..."

It is found that nuclei with even numbers of protons and neutrons are more stable than those with odd numbers. In particular, there are "magic numbers" of neutrons and protons which seem to be particularly favored in terms of nuclear stability:

Nuclei with either numbers of protons or neutrons equal to Z,N = 2,8,20,28,50,82 or 126 exhibit certain properties including anomalously low masses, high natural abundances and high energy first excited states.
Calcium provides a good example of the exceptional stability of "doubly magic" nuclei since it has two of them. The existence of several stable isotopes of calcium may have to to with the fact that Z=20, a magic number. The two highlighted isotopes have neutron numbers 20 and 28, also magic numbers. The existence of these magic numbers suggests closed shell configurations, like the shells in atomic structure. A shell model is one in which the system is thought to consist of individual particles moving in bound orbitals in response to the remainder of the system. Each orbital has a well designated energy, angular momentum, and parity associated with it.

Numbers, known as figurate or polygonal numbers, appeared in 15th-century arithmetic books and were probably known to the ancient Chinese; but they were of especial interest to the ancient Greek mathematicians. To the Pythagoreans (c. 500 BC), numbers were of paramount significance; everything could be explained by numbers, and numbers were invested with specific characteristics and personalities. Pascal`s Triangle is an arrangement of numbers such that each number is the sum of two numbers immediately above it in the previous row.Pascal simply discovered one of nature`s structural formulas.

Pascal`s Triangle is the basic number formula in Nature. The natural numbers, triangular numbers and tetrahedral numbers can be found as column#2, column#3 and column#4 in Pascal`s Triangle :

There is a connection between tetrahedral numbers and triangular numbers and the magic numbers , as shown in the table below :

There is a remarkable connection between magic numbers on one side and pyramidal numbers, tetrahedral numbers and triangular numbers, on the other :

Here is the alternative form of this connection, as shown in the table below:

The chemistry of the atom depends mainly on the electrons and only indirectly upon the nucleus, which supplies the positive electrostatic charge to attract the electrons. The model for the behavior of the electrons seems to be well understood. However, the structure and behavior of the nucleus is less well understood. Several competing models have been proposed. The Liquid Drop Model of the nucleus (Niels Bohr, 1936) compares the nuclear forces to molecular forces acting in a liquid drop. In 1939 Niels Bohr and John Wheeler used the Liquid Drop Model to explain the process of nuclear fission. The model proposed in this paper is also used to explain nuclear fission but in a more comprehensive and precise manner. The Liquid Drop Model fails to explain the unusual properties associated with the magic number nuclei. Another model called the Shell Model (Mayer, 1949) is very different from the liquid drop model. In the Shell model, each nucleon moves in a well defined orbit within the nucleus and hardly makes any collisions at all. This model is similar to the quantum model for electrons. The Pauli exclusion principle applies to the nucleus as well as to electrons according to the Shell model. The Shell Model helped to explain many nuclear properties and offered one explanation for the magic numbers.

The next table expresses finite differences of the magic numbers :

We suggest the original series of the "magic numbers" of neutrons and protons in accordance with the Pascal Triangle law:

For magic numbers 2, 6, 20, 28, 50, 82, 126 the geometrical image is offered:

There is a connection between tetrahedral numbers and triangular numbers and the "new" magic numbers , as shown in the table below :

The next table expresses finite differences of the "new" magic numbers :

Magic numbers are featured by the formula:

See: For m=8 ZN=184.

The magic numbers are used by as the basis for a new theoretical model of the atomic nucleus.( see Shell Model of Nucleus ). The magic numbers are reduced to a series of natural numbers and sums of natural numbers, which are utilized to reconstruct the magic numbers and to produce a Periodic Table of the Nucleus of all elements with respect to protons...

In the atomic shell model, the shells are filled with protons in order of increasing natural numbers series, producing the magic numbers. These elements have highly stable properties:

Magic numbers of protons are given in form of three-dimensional Pascal`s triangle

See also a triangles form of numbers