In this paper we will deal more thoroughly with the prime numbers in biology. The prime numbers are integer positive numbers divisible only by themselves and the unit 1, eg 2, 3, 5, 7, 11, 13,17, 19 … The primes that differ by 2 are called twin primes. Euclid was the first who proved that the number of primes is infinite. Today we know that the greater the interval between the numbers is, the more prime sparsity we have. On the other hand, complex numbers (all positive integers that are not primes) result as a raw product of primes, e.g. 70 = 2 x 5 x 7 etc. Therefore any positive integer greater than 1 is either a prime or product of primes. Mathematicians for their own reasons, consider the number 1 neither a prime nor a complex number. Prime numbers, are for Mathematics what for Chemistry and Physics are atoms of matter: the building blocks upon which all other numbers are built, the ”atoms” that compose the Mathematics’ universe. Someone said that mathematicians love prime numbers, such as chemists love atoms and biologists love genes. The interest in prime numbers is big and despite the era, it remains at the forefront of scientific activity. Moreover, there are major unresolved problems over time regarding prime numbers (Ryman conjecture, Goldbach conjecture, conjecture of the twin primes etc.). Also, there are no mathematical formulas known to compute the next prime number, and how many prime numbers are smaller than a specific number. Two very characteristic sayings of great scientists about the importance of the above are the following: Hilbert, a great German mathematician had said that ”if it was possible after 1000 years from my death to come back to life, the first thing I would ask would be if Ryman’s conjecture has been resolved. ” This said, knowing, as all mathematicians, that the problem is extremely difficult and the solution of it will lead to numerous solutions of other mathematical problems.

Paul Erdős, a hungarian, jewish great mathematician who dealt with prime numbers, just before his death said: ”It will take one million years until we understand prime numbers.”

Let’s have a look at primes’ relations with biological issues, starting with some simple observations:

a) The codons of the genetic code that encrypt amino acids are 61, while there are three (dating) that do not encrypt any amino acid. Each codon consists of three nucleotides.

b) Among the 37 genes of mitochondrial DNA 13 encode proteins.

c) The smallest form of life, a bacterium that belongs to the mycoplasma has 521 genes (a prime number).

d) To nucleic acids (DNA, RNA), the orientation of the polynucleotide chain is 5’ → 3’.

e) Microtubules, which are structures of the cellular shell of the ciliums and flagellums of various eukaryotic cells, consist of 13 parallel protofibrils (with a 5nm diameter each) of alternating molecules of alpha and beta tubulin.

f) 2 and 3 phosphate groups of nucleotides (ADP, ATP, etc.), 2, 3, 5, N atoms depending on the nitrogenous base of nucleotides, consist of 5 C atoms therein. g) 5radial symmetry in echinoderms, e.g. starfish. h) 23 chromosomes in human reproductive cell etc. i) In the genome of prokaryotic organisms, the genes for enzymes involved in a metabolic way, are organized in operons, i.e. groups that are subject to common control of their expression. The lactose operon has three structural genes, many other 2 or 3 structural genes, tryptophan’s has 5 etc.

In order to avoid any misunderstanding, we clarify that in many biological issues numerical quantities involved, are not prime numbers, but can be complex or decimal numbers etc. However, our interest in this paper focuses on prime numbers and their correlation with biological issues. We continue with examples of prime numbers that appear in natural selection issues and in evolution of organisms. A known example from the field of the animal kingdom is that two kinds of cicadas Magicicada tredecim and Magicicada septendecim live in the same environment with a lifecycle of 13 and 17 years respectively. Throughout their life, except the last year, they live in soil and feed on the juices of tree roots. During the last year of their life, for a few weeks, they transform from larvae to adult, occupy the forest, eat, lay eggs and die. But how these species evolved so that their cycle of life to be prime numbers? One answer is that their mutual emergence to the forest takes too long and happens once in 13 x 17 = 221 years. While if their life cycles were complex numbers, eg 12 and 18 years in the above time they would have competed six times, as many are the common multiples of 12 and 18. As we can see, the prime numbers 13 and 17 are not something abstract and random, but the basis of their survival. Such examples are likely to exist in animal and plant kingdom.

The birthday paradox is very interesting. It belongs to the probability theory and refers to a problem which by the common sense has an unlikely answer. The formulation of the problem is this: In a group of 23 people which is the possibility that two among these individuals have their birthdays on the same day? The obvious answer is 23/365 = 0.063, i.e. 6%. Despite this, the mathematical solution is 50%. The possibility is even 100% when it refers to 367 people, including those who were born on February 29. 23 and 367 are prime numbers, and the reference to this paradox is used because it is related to people (biological beings) and a biological fact, their birth in a specific time period (birthday). For total of 23 peopleA typical example are the two players football teams and the referee in a fight. It is worth mentioning that this paradox is the basis for one of the most common methods of cryptonalysis, in the corresponding field of computer science (cryptography).

Prime numbers are related to the molecular basis of apoptosis, or programmed cell death, which was initially studied in the filamentary worm Caenorhabditis elegans. Later, it was studied in other invertebrates and vertebrates. Only in the hermaphrodite filamentary worm, we have the best of apoptosis study system, to the point that is considered to be created for this purpose. Apoptosis is a fundamental biological process that is necessary for the removal of surplus cells during embryonic development, maintaining the homeostasis in the adult organism, maturation of cell populations of the hematopoietic and immune system and malignant transformation.

Typically the apoptosis is not a consequence of disfunction and therefore differs from necrosis, which is always a result of a harmful effect. In filamentary worm, during its development when 1090 cells are produced (hence 1091 total with the original zygote), of which 131 somatic cells are eliminated by the process of programmed cell death while 3 genes are responsible for it. 1091, 131 and 3 are prime numbers, and the above constitute an ascertainment, the causes of which are unknown for now. Maybe in the future for the entire process of the phenomenon, Emerged and a mathematical model.