We introduce a series of
Z
2
n
\mathbb {Z}_2^n
-graded quasialgebras
P
n
(
m
)
\mathbb {P}_n(m)
which generalizes Clifford algebras, higher octonions, and higher Cayley algebras. The constructed series of algebras and their minor perturbations are applied to contribute explicit solutions to the Hurwitz problem on compositions of quadratic forms. In particular, we provide explicit expressions of the well-known Hurwitz-Radon square identities in a uniform way, recover the Yuzvinsky-Lam-Smith formulas, confirm the third family of admissible triples proposed by Yuzvinsky in 1984, improve the two infinite families of solutions obtained recently by Lenzhen, Morier-Genoud and Ovsienko, and construct several new infinite families of solutions.
Kamke [Kam21] solved an analog of Waring's problem with nth powers replaced by integer-valued polynomials. Larsen and Nguyen [LN19] explored the view of algebraic groups as a natural setting for Waring's problem. This paper generalizes integer-valued polynomials to polynomial maps from nonempty commutative semigroups to arbitrary groups, in particular, to polynomial sequences from nonnegative integers to locally nilpotent groups. The main result is an analog of Waring's problem for the general discrete Heisenberg groups H2n+1(Z) for any n ≥ 1.
This paper develops a theory of polynomial maps from commutative semigroups to arbitrary groups and proves that it has desirable formal properties when the target group is locally nilpotent. We will apply this theory to solve Waring's problem for general discrete Heisenberg groups in a sequel to this paper.
The purpose of this paper is to introduce the concept of reflecting numbers to the realm of number theory and to classify reflecting numbers of certain types. For us, reflecting numbers are coming from congruent numbers, above congruent numbers, and away from congruent numbers.Explicitly speaking, a reflecting number of type (k, m) is the average of two distinct rational kth powers, between which the distance is twice another nonzero rational mth power. In particular, reflecting numbers of type (2, 2) are all congruent numbers and thus will be called reflecting congruent numbers in this paper. We can show that all prime numbers p ≡ 5 mod 8 are reflecting congruent and in general for any integer k ≥ 0 there are infinitely many squarefree reflecting congruent numbers in the residue class of 5 modulo 8 with exactly k + 1 prime divisors. Moreover, we conjecture that all prime congruent numbers p ≡ 1 mod 8 are reflecting congruent. In addition, we show that there are no reflecting numbers of type (k, m) if gcd(k, m) ≥ 3.
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