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According to Carl Friedrich Gauss (1777–1855), mathematics is the queen of the sciences—and number theory is the queen of mathematics. Numbers (integers, algebraic integers, transcendental numbers, p-adic numbers) and symmetries are investigated in the nine refereed papers of this MDPI issue. This book shows how symmetry pervades number theory. In particular, it highlights connections between symmetry and number theory, quantum computing and elementary particles (thanks to 3-manifolds), and other branches of mathematics (such as probability spaces) and revisits standard subjects (such as the Sieve procedure, primality tests, and Pascal’s triangle). The book should be of interest to all mathematicians, and physicists.

quantum computation --- IC-POVMs --- knot theory --- three-manifolds --- branch coverings --- Dehn surgeries --- zeta function --- Pólya-Hilbert conjecture --- Riemann interferometer --- prime numbers --- Prime Number Theorem (P.N.T.) --- modified Sieve procedure --- binary periodical sequences --- prime number function --- prime characteristic function --- limited intervals --- logarithmic integral estimations --- twin prime numbers --- free probability --- p-adic number fields ℚp --- Banach ∗-probability spaces --- C*-algebras --- semicircular elements --- the semicircular law --- asymptotic semicircular laws --- Kaprekar constants --- Kaprekar transformation --- fixed points for recursive functions --- Baker’s theorem --- Gel’fond–Schneider theorem --- algebraic number --- transcendental number --- standard model of elementary particles --- 4-manifold topology --- particles as 3-Braids --- branched coverings --- knots and links --- charge as Hirzebruch defect --- umbral moonshine --- number of generations --- the pe-Pascal’s triangle --- Lucas’ result on the Pascal’s triangle --- congruences of binomial expansions --- primality test --- Miller–Rabin primality test --- strong pseudoprimes --- primality witnesses