Bang's Theorem (other)

	Bang's Theorem (other) Bang's theorem may refer to: * The solution to Tarski's plank problem by Thøger Bang (1917–1997) * A special case of Zsigmondy's theorem In number theory, Zsigmondy's theorem, named after Karl Zsigmondy, states that if a>b>0 are coprime integers, then for any integer n \ge 1, there is a prime number ''p'' (called a ''primitive prime divisor'') that divides a^n-b^n and does not divi ..., on unique divisors of Mersenne numbers, by Alfred Sophus Bang (1866–1942) * Bang's theorem on tetrahedra, posed by Alfred Sophus Bang (1866–1942) {{dab ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Tarski's Plank Problem In mathematics, Tarski's plank problem is a question about coverings of convex regions in ''n''-dimensional Euclidean space by "planks": regions between two hyperplanes. Tarski asked if the sum of the widths of the planks must be at least the minimum width of the convex region. The question was answered affirmatively by . Statement Given a convex body ''C'' in R''n'' and a hyperplane ''H'', the width of ''C'' parallel to ''H'', ''w''(''C'',''H''), is the distance between the two supporting hyperplanes of ''C'' that are parallel to ''H''. The smallest such distance (i.e. the infimum over all possible hyperplanes) is called the minimal width of ''C'', ''w''(''C''). The (closed) set of points ''P'' between two distinct, parallel hyperplanes in R''n'' is called a plank, and the distance between the two hyperplanes is called the width of the plank, ''w''(''P''). Tarski conjectured that if a convex body ''C'' of minimal width ''w''(''C'') was covered by a collection of planks, then ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Zsigmondy's Theorem In number theory, Zsigmondy's theorem, named after Karl Zsigmondy, states that if a>b>0 are coprime integers, then for any integer n \ge 1, there is a prime number ''p'' (called a ''primitive prime divisor'') that divides a^n-b^n and does not divide a^k-b^k for any positive integer k1 and n is not equal to 6, then 2^n-1 has a prime divisor not dividing any 2^k-1 with k. Similarly, $a^n+b^n$ has at least one primitive prime divisor with the exception $2^3+1^3=9$ . Zsigmondy's theorem is often useful, especially in group theory, where it is used to prove that various groups have distinct orders except when they are known to be the same. History The theorem was discovered by Zsigmondy working in Vienna from 1894 until 1925. Generalizations Let $(a_n)_$ be a sequence of nonzero integers. The Zs ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]

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