Prime number formula

Prime numbers do not occur in a predictable way. There are sequences of primes which can be partially described in a formula, but sooner or later the formula breaks down. One formula, invented by Marin Mersenne (1588-1648) is 2P - 1, where p is a prime number. Although this formula generates many primes, it also misses many primes. Another formula, invented by Leonhard Euler (1707-1783), generates prime numbers regularly for the series of consecutive numbers from 0 to 15 and then stops. The formula is + x + 17, in which x is any number from 0 to 15. 

One of Fermat s many theorems provides a quick way of finding out if a number is prime. Say n is any whole number, and p is any prime number. Raise n to the power of p, and then subtract n from the result. If p is really a prime number, then the result can be divided evenly by p. If anything is left over after the division, then the number p is not prime. A shorter way of putting this formula is this nP - n can be divided evenly by p. 

Many mathematicians, including Mersenne and Euler, have tried to find a formula that will define all the prime numbers. No one has ever succeeded. 

Fermat had one of the most famous failures. He thought that if he squared 2 and then raised the square of 2 to a higher power, which he labeled n (a whole number), then the results would be nothing but primes. His formula looks like this 22" + 1 = a prime number. This formula appeared to work until Leonhard Euler proved it wrong. Euler found that if 5 is substituted for n in the formula 22n + 1, the resulting number is 4,294,967,297, which can be divided equally by 641 and 6,700,417. 

RoSc62 J. B. Rosser, L. Schoenfield Approximate Formulas for Some Functions of Prime Numbers Illinois J. Math. 6 (1962) 64-94. 

This formula gives 26 prime numbers, for all n from 0 to 25. 

If we adopt the system of numbering each compd separately, and if the compd on the right side of the formula has simple arabic numerals (1,2,3, etc) counted clockwise, while the compd on the left side has primed arabic numerals 1, 2,3 etc, counted "counter-clockwise", then the formula may be represented as ... 

In rare cases in which a third ring compound is attached to one of the intermediate N atoms of an open chain nitrogen compound, its substituents would be numbered clockwise using double primes, as 1",2 ",3 , etc. In some cases the groups NH2, N02, etc may be attached to intermediate N atoms of the open N chain. Following is the formula of a complicated hypothetical compound ... 

Number theory is full of famous formulas that illustrate the relationships between whole numbers from 1 to infinity. Some of these formulas are very complicated, but the most famous ones are very simple, for example, the theorem by Fermat below that proves if a number is prime. 

If we adopt as a rule that any compd numbered from the left of the formula with numerals marked with a prime ( ) sign (such as , 2, etc) with the numbering done counter-clockwise, and any ring compd on the right hand side of the formula carrying plain numerals, such as 1,2 etc, and the numbering is done clockwise, the design-nation will be simplified... 

Fig. 3. Structural formula of tetraphenylene 1 showing the numbering system used and labelling of the chemically equivalent bond lengths, bond angles, and torsion angles. Atoms labelled with primes are related to the reference atoms by two-fold rotational symmetry...

But careful study of his myriad formulas leaves no doubt that he was most strongly influenced by Gerhardtian molecular positivism. His formulas were constructed to create an equal number of prime marks on both sides of the bracket, not to express presumed sequential physical arrangements of atoms within the molecule—much less any presumed linking function of atoms or radicals. For example, for hyposulfuric acid, sodium hyposulfite, anhydrous carbonic and sulfuric acids, and metaphosphoric acid, he wrote ... 

By searching other key fields, such as the molecular formula, the CAS registry number or the molecular, structure-related Beilstein Registry Number (Beilstein Prime Key). 

The situation becomes more complicated as more elements and fewer limitations of their number must be taken into account. In practice, one must try to restrict oneself to certain elements and a maximum and/or minimum number of certain isotopes to assure a high degree of confidence in the assignment of formulas. Isotopic patterns provide a prime source of such additional information. Combining the information from accurate mass data and experimental peak intensities with calculated isotopic patterns allows to significantly reduce the number of potential elemental compositions of a particular ion [46,47]. 

In speaking of A, or using A, an index of eigenfunctions, we understand A always as the complete symbol of (2), or (2a), (2a ). If A appears as a number in a formula instead, it represents only the numerical value of A, which is obtained when we omit the prime [Komma] and the + and —. 

Extended Molecular Formula (EMF). This search key consists of the molecular formula of each fragment in a compound, qualified by the number of prime rings in that moleform and the number of direct non-hydrogen atoms attached to rings. 

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