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1 - one

1 (one) is usually categorized as a numeral. In some languages, however, “one” acts like an adjective in the sense that it agrees with noun in gender. For instance, in Hebrew it follows the noun and agrees  איש אחד (one man) and אשה אחת (one woman). Similarly, in Slavonic languages they occur before the noun  and can be declined according to gender and case. For instance, we have “jeden muž” (one man) and “jedna žena” (one woman), “jedno okno” (one window) in Czech or Slovak. The number “one” also forms plural in these languages: אחדים  in Hebrew,  or “jedni” (masculine form) or  “jedny” (feminine or neuter form) in Czech and Slovak.  In Greek, even the cardinal numbers for one ευα  (m: ευας, f: μια, n: ευα), two (δυο), three τρια  (m: τρεις, f: τρεις, n: τρια), and four τεσσερα (m: τεσσερια, f: τεσσερις, n: τεσσερα), have the form of adjectives and  they can be declined according to gender and case.

Euclid (~300 B.C)  in Definition 1 of Book VII of his Elements  says: A  unit  is that by virtue of which each  of the things that exist is called one.  Number 1 has a special position and Euclid treats the unit, 1, separately from the numbers.  Euclid was not alone who held this point of view.  Aristotle (384 - 322 B.C.) in his Metaphysics ( [1]  , p. 68) wrote that One is reasonably regarded as not being itself a number, because a measure is not the things measured, but the measure or the One is the beginning (or principle) of number.  Health expresses the opinion that this doctrine may be of Pythagorean origin because it can be found by Nicomachus in his Introduction to Arithmetic, an influential treatise on number theory which is considered as a standard reference of the pronouncement of Pythagorean school on elementary theory and properties of numbers.

Thymaridas of Paros (400-350 B.C.), a follower  Pythagoras,  called  'one' a  ‘limiting quantity' or a 'limit of fewness'.

The Greek identified various properties of different numbers with gods. The monad    ,  the primary element of the number one ,  was identified with the primordial chaos, which existed before the gods.  It was also identified with the sun and with Apollo, the god of the sun.  Nicomachus held the number ‘One’ to be itself the Divinity, Reason, the Principle of form and goodness. For Moderatus of Gades the number One was the symbol of unity and similarity, the principle of harmony and of the constitution of all things.

Whether 1 is or is not a prime is a matter of definition.

In Definition 11 of Book VII of his Elements  defines: A prime number is that which is measured by a unit alone.  Thymaridas who wrote on prime numbers called a prime number rectilinear since it can only be represented one-dimensionally.

Most modern textbooks classify number 1 neither prime nor composite, but older texts generally considered it to be prime. In 1859, V.A.Le Besgue (written also as Lebesgue) stated explicitly that 1 is prime in his Exercices d'analyse numérique. 1 is a prime also in Primary Elements of Algebra for Common Schools and Academies (1866) by Joseph Ray and Standard Arithmetic (1892) by William J. Milne.

Henri Lebesgue ¹  (1875-1941) is said to be the last professional mathematician who classified 1 as a prime in 1899.  But this is not completely true, because even later  some number theorists did list "1" as a prime, e.g.  D.N. Lehmer did so in his list [2]  of primes to 10,006,721 published in 1914. It is almost certain that he done this  from historical reasons.  The most prominent person among the non-professional mathematicians who considered 1 as a prime  was probably C. Sagan in his novel Contact (p.86) in 1985.

The reason not to accept 1 as a prime, is attributed to the so-called fundamental theorem of arithmetic which says that “each number has a unique factorization into primes”.  If 1 is accepted as a prime this statement is not valid in the mentioned form. Though this result was tacitly used long before, it was as such was explicitly  stated for the first time by C.F.Gauß , but as an irony, Moritz Abraham Stern (1807-1894) who succeeded Gauß as Ordinarius (full professor) at Göttingen University in 1858  still considered 1 as prime number.

As an example, a Stern prime is a prime number that is not the sum of a smaller prime and twice the square of a nonzero integer. Stern was motivated to study such primes by a conjecture formulated by Goldbach in a letter to  Euler claiming that every odd integer is of the form where   is an integer, including zero.  Since Stern considered 1 as a prime, 3 is not a Stern prime because it could be represented as (this is, however, the only difference between Stern’s list and our list).  

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