Starting from the difficulty of creating playful representation of domain-specific abstract concepts, this study discusses the design of Prime Slaughter, a computer game aimed at facilitating individual sense-making of abstract mathematical concepts. Specifically the game proposes a transpositionmathematical concepts. Specifically the game proposes a transposition of primality and factorization...

In Ok, the rational prime 2 has the principal prime factorization (2) = ( 3 √ 2)3, so k has class number 1: the ring Ok has unique factorization. Next we show Ok = Z[ 3 √ 2]. Let α = a+b 3 √ 2+c 3 √ 4 be an algebraic integer, with a, b, c all rational. Computing Trk/Q of α, α 3 √ 2, and α 3 √ 4 we see 3a, 6b, 6c ∈ Z. So the denominators of a, b, and c involve at most 2 and 3. To show 2 and 3 do...

In a commutative f-ring, an 1-ideal I is called pseudoprime if ab = 0 implies a E I or b E I, and is called square dominated if for every a E I, lal < x2 for some x E A such that x2 E I. Several characterizations of pseudoprime 1-ideals are given in the class of commutative semiprime frings in which minimal prime 1-ideals are square dominated. It is shown that the hypothesis imposed on the f-ri...

Let \((a, b, c)\) be a primitive Pythagorean triple parameterized as \(a=u^2-v^2, b=2uv, c=u^2+v^2\), where \(u&gt;v&gt;0\) are co-prime and not of the same parity. In 1956, L. Jesmanowicz conjectured that for any positive integer \(n\), Diophantine equation \((an)^x+(bn)^y=(cn)^z\) has only solution \((x,y,z)=(2,2,2)\). this connection we call \((x,y,z)\ne (2,2,2)\) with \(n&gt;1\) exceptional...

We prove, without assuming the Generalized Riemann Hypothesis, but with at most one exception, that an irreducible cyclic code c(p, m, v) with v prime and p of index 2 modulo v is a two-weight code if and only if it is a semiprimitive code or it is one of the six sporadic known codes. The result is proved without any exception for index-two irreducible cyclic c(p, m, v) codes with v prime not c...

Let Q(J!k) be an imaginary quadratic "eld with discriminant !k and class number h, with kO3, 4, or 8. Let p be a prime such that (~k p )"1. There are integers C, D, unique up to sign, such that 4ph"C2#kD2, p PC. Stickelberger gave a congruence for C modulo p which extends congruences of Gauss, Jacobi, and Eisenstein. Stickelberger also gave a simultaneous congruence for C modulo k, but only for...

Abstract A new scheme of state classification is proposed and applied to analyze masses the heavy baryons $$\varOmega _{Q}$$ Ω Q , $$\varSigma Σ $$\varXi _{Q}^{\prime }$$ Ξ ′ </mml:m...

Throughout this paper, every groups are finite. The prime graph of a group $G$ is denoted by $Gamma(G)$. Also $G$ is called recognizable by prime graph if for every finite group $H$ with $Gamma(H) = Gamma(G)$, we conclude that $Gcong H$. Until now, it is proved that if $k$ is an odd number and $p$ is an odd prime number, then $PGL(2,p^k)$ is recognizable by prime graph. So if $k$ is even, the r...