arXiv subjects
Publications and source records attributed to Constantin M. Petridi.
We examine the sums $S(k,\,n)$ of the $k-$th powers of the $ϕ(n)$ integers $α_1<α_2<\cdots<α_{ϕ(n)}$ less than and prime to $n$ (Euler set) and prove a formula (new) for $S(3,\,n)$. If $n$ equals a prime $p$, we prove a theorem showing a connection of $S(3,\,p)$ with Artin's conjectural constant for primitive roots and with other functions involving primes.
An extended generating series of the radical of n, involving two variables, leads to an identity in said variables, which proves Bombieri's abc-Conjecture for certain sets of integers.
We extend the theory (formal part only} of algebras with one binary operation (our paper arXiv:math/0110333v1 [math.RA] 31 Oct 2001) to algebras with several operations of any arity.
We prove that for a positive integer a the integer sequence P(n) satisfying for all n, -infty<n<infty, the recurrence P(n)=a+P(n-phi(a)), phi(a) the Euler function, generates in increasing order all integers P(n) coprime to a.The finite Fourier expansion of P(n) is given in terms of a, n, and the phi(a)-th roots of unity. Properties of the sequence are derived.
We classify integer abc-equations c = a + b (to be defined), according to their radical R(abc) and prove that the resulting equivalence classes contain only a finite number of such equations. The proof depends on a 1933 theorem of Kurt Mahler.
In our paper arXiv: math.RA/0110333 v1 Oct 2001 we showed that the number of algebras defined by a binary operation satisfying a formally irreducible identity between two n-iterates is O( e^{-n/16}S_{n}^{2} for n --> infinity, S_{n} being the nth-Catalan number. This was proved by using exclusively the series of tableaux A_{n}. By using also the series of tableaux B_{n}, we now sharpen this result to O{(n+2)/n|e^{-n/16}-2/n)|S_{n}^{2}}
We prove that for a positive integer $c$ and any given $\varepsilon$, $0<\varepsilon<1$, the number $N(c)$ of equations $c=a+b$, $a<b$, with positive coprime integers $a$ and $b$, which satisfy the inequality $$c < R(c)^{\frac{\varepsilon}{1+\varepsilon}}R(a)^{\frac{1}{1+\varepsilon}}R(b)^{\frac{1}{1+\varepsilon}},$$ where R(n) is the radical of $n$, is for $c\to\infty$ $$N(c)=(1-\varepsilon)\frac{ϕ(c)}{2}+O\Bigl(\frac{ϕ(c)}{2}\Bigr).$$ An analogue for the abc-conjecture inequality $c<R(abc)^{1+\varepsilon}$ (without a constant factor) will also be proved.
We prove that for any given s > 0 the geometric mean of the radicals of the phi(n) integers less than n and relatively prime to n, is greater than K(s)R(n) ^(-s)n, where K(s) an absolute constant depending only on s.
We prove that for any positive integer c and any s > 0 there are representations of c as a sum a+b of two coprime positive integers a, b, such that the respective radicals are all greater than K(s)R(c)^(1-s)c^2. For the reprasentations in question, this is a stronger result than the abc-conjecture, which postulates that these representations are greater than k(s)c^1/(1+s).
We prove that for any positive integer c there are at least N(c), $1\leq N(c) < ϕ(c)/2$ representations of c as a sum of two positive integers a, b, with no common divisor, such that the N(c) radicals R(abc) are all greater than kc, where k an absolute constant.
We reduce the principal problem of Additive Number Theory of whether an infinite sequence of integers constitutes a finite basis for the integers to a Diophantine problem involving the difference set of the sequence, by proving a formula connecting the respective number of solutions.
We propound the thesis that there is a limitation to the number of possible structures which are axiomatically endowed with identities involving operations. In the case of algebras with a binary operation satisfying a formally reducible (to be explained) identity between two n-iterates of the operation, it is established that the frequency of such algebras goes to zero as n tends to infinity. This is proved by a suitable ordering and labeling of the expressions (words) of the corresponding free algebra and the formation of a series of tableaux whose entries are the labels. The tableaux reveal surprising symmetry properties, stated in terms of the Catalan numbers and their partitions. As a result of the defining identity and the tableaux all iterates of order higher than n fall into equivalence classes of semantically equal ones. Classnumbers depending on the tableaux are calculated for all algebras of order n = 3 (and partially for n = 4). Certain classnumbers are invariants in the sense that for algebras of same order they are equal. Algebras with several operations of any arity are considered. A generalization of Catalan numbers depending on homorphisms of the structure is proposed and corresponding generating functions set up. As an example of this, a kind of skein polynomials are constructed characterizing the formal build-up of iterates. As no distinction is made between the various algebras, isomorphic or not, which are models of the identity the results can also be formulated in terms of varieties with signature the identity in question.