## What is/are Many Prime?

Many Prime - We study the natural density of the set of primes of K having some prescribed Frobenius symbol in \( \operatorname {\mathrm {Gal}}(L/K)\), and for which the reduction of G has multiplicative order with some prescribed l-adic valuation for finitely many prime numbers l.^{[1]}Two other applications we will meet are a proof by calculus that there are infinitely many primes and a proof that e is irrational.

^{[2]}There are many primeval methods for Internet Traffic Classification like port and payload based classifiers.

^{[3]}Furthermore, strong abelian groups are characterised to be precisely those abelian groups A such that there are only finitely many primes p such that the group A / pA is infinite and for every prime p , there are only finitely many natural numbers n such that $\left( {p^n A} \right)[p]/\left( {p^{n + 1} A} \right)[p]$ is infinite.

^{[4]}In commutative Krull monoids which have sufficiently many prime divisors in all classes of their class group, the set of catenary degrees and the set of tame degrees are intervals.

^{[5]}Our treatment builds on recent work by Salberger which brings in many primes in Heath-Brown's variant of the determinant method, and on recent work by Walsh and Ellenberg--Venkatesh, who bring in the size of the defining polynomial.

^{[6]}The target miRNA sequences can bind two ssDNA probes to form a junction structure to initiate a dual polymerization/nicking cyclic reaction for the production of many primers, which further trigger multiple RCA reactions in a drastically amplified sequence replication and extension mode for the yield of substantial dsDNAs with various sizes.

^{[7]}Thus, it must deter potential entry ex ante by preempting many prime product and geographic locations.

^{[8]}We construct several new families of non-congruent numbers with arbitrarily many prime factors congruent to 3 modulo 8.

^{[9]}Although many Prime Screen items correlated with mental well-being as expected, correlations between item scores and mental well-being were non-significant for poor fit items.

^{[10]}We establish the main conjecture of Iwasawa theory for the Iwasawa module $X(H_\infty)$ defined for a non-cyclotomic $\mathbb{Z}_p$-extension $H_\infty/H$, where $H$ is the Hilbert class field of an imaginary quadratic field $K$, at infinitely many primes $p$ which split in $K$ including $p=2$.

^{[11]}We show that if a big set $S\subseteq Z$ of rational points of bounded height occupies few residue classes modulo $\mathfrak{p}$ for many prime ideals $\mathfrak{p}$, then a positive proportion of $S$ must lie in the zero set of a polynomial of low degree that does not vanish at $Z$.

^{[12]}We support this conjecture by checking all the cases for many primes.

^{[13]}This paper gives some new families of non-congruent numbers with arbitrarily many prime divisors.

^{[14]}To extend Iwasawa's classical theorem from Z p -towers to Z p d -towers, Greenberg conjectured that the exponent of p in the n-th class number in a Z p d -tower of a global field K ramified at finitely many primes is given by a polynomial in p n and n of total degree at most d for sufficiently large n.

^{[15]}Using these methods, we prove that there are infinitely many primes p such that $$\mathrm{PG}(4,p)$$PG(4,p) contains a $$\mathrm{PSL}(2,11)$$PSL(2,11)-invariant 110-arc, where $$\mathrm{PSL}(2,11)$$PSL(2,11) is given in one of its natural irreducible representations as a subgroup of $$\mathrm{PGL}(5,p)$$PGL(5,p).

^{[16]}We show that there are infinitely many primes $p$ such that $p-1$ is divisible by a square $d^2 \geq p^\theta$ for $\theta=1/2+1/2000.

^{[17]}This article shows the existence of superspecial and maximal nonhyperelliptic curves of genera four and five in characteristic p > 0 for infinitely many primes p.

^{[18]}Facing tradition against contemporary ideas is now a challenge in many primeval countries.

^{[19]}

## Infinitely Many Prime

Two other applications we will meet are a proof by calculus that there are infinitely many primes and a proof that e is irrational.^{[1]}We establish the main conjecture of Iwasawa theory for the Iwasawa module $X(H_\infty)$ defined for a non-cyclotomic $\mathbb{Z}_p$-extension $H_\infty/H$, where $H$ is the Hilbert class field of an imaginary quadratic field $K$, at infinitely many primes $p$ which split in $K$ including $p=2$.

^{[2]}Using these methods, we prove that there are infinitely many primes p such that $$\mathrm{PG}(4,p)$$PG(4,p) contains a $$\mathrm{PSL}(2,11)$$PSL(2,11)-invariant 110-arc, where $$\mathrm{PSL}(2,11)$$PSL(2,11) is given in one of its natural irreducible representations as a subgroup of $$\mathrm{PGL}(5,p)$$PGL(5,p).

^{[3]}We show that there are infinitely many primes $p$ such that $p-1$ is divisible by a square $d^2 \geq p^\theta$ for $\theta=1/2+1/2000.

^{[4]}This article shows the existence of superspecial and maximal nonhyperelliptic curves of genera four and five in characteristic p > 0 for infinitely many primes p.

^{[5]}

## Finitely Many Prime

We study the natural density of the set of primes of K having some prescribed Frobenius symbol in \( \operatorname {\mathrm {Gal}}(L/K)\), and for which the reduction of G has multiplicative order with some prescribed l-adic valuation for finitely many prime numbers l.^{[1]}Furthermore, strong abelian groups are characterised to be precisely those abelian groups A such that there are only finitely many primes p such that the group A / pA is infinite and for every prime p , there are only finitely many natural numbers n such that $\left( {p^n A} \right)[p]/\left( {p^{n + 1} A} \right)[p]$ is infinite.

^{[2]}To extend Iwasawa's classical theorem from Z p -towers to Z p d -towers, Greenberg conjectured that the exponent of p in the n-th class number in a Z p d -tower of a global field K ramified at finitely many primes is given by a polynomial in p n and n of total degree at most d for sufficiently large n.

^{[3]}

## Arbitrarily Many Prime

We construct several new families of non-congruent numbers with arbitrarily many prime factors congruent to 3 modulo 8.^{[1]}This paper gives some new families of non-congruent numbers with arbitrarily many prime divisors.

^{[2]}

## many prime divisor

In commutative Krull monoids which have sufficiently many prime divisors in all classes of their class group, the set of catenary degrees and the set of tame degrees are intervals.^{[1]}This paper gives some new families of non-congruent numbers with arbitrarily many prime divisors.

^{[2]}