Information AboutSquare-free Integer |
| CATEGORIES ABOUT SQUARE-FREE INTEGER | |
| number theory | |
| integer sequences | |
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: 1 , 2 , 3 , 5 , 6 , 7 , 10 , 11 , 13 , 14 , 15 , 17 , 19 , 21 , 22 , 23 , 26 , 29 , 30 , 31 , 33 , ... (sequence in OEIS ) EQUIVALENT CHARACTERIZATIONS OF SQUARE-FREE NUMBERS The integer ''n'' is square-free if and only if in the . The positive integer ''n'' is square-free If And Only If μ(''n'') ≠ 0, where μ denotes the Möbius Function . The positive integer ''n'' is square-free if and only if all Abelian Group s of Order ''n'' are Isomorphic , which is the case if and only if all of them are Cyclic . This follows from the classification of Finitely Generated Abelian Group s. The integer ''n'' is square-free iff the Factor Ring Z / nZ (see Modular Arithmetic ) is a Product of Field s. This follows from the Chinese Remainder Theorem and the fact that a ring of the form Z / kZ is a field if and only if ''k'' is a prime. For every positive integer ''n'', the set of all positive divisors of ''n'' becomes a Partially Ordered Set if we use Divisibility as the order relation. This partially ordered set is always a Distributive Lattice . It is a Boolean Algebra if and only if ''n'' is square-free. Given the positive integer ''n'', define the radical of the integer ''n'' by m equal to the product of the prime numbers ''p'' dividing ''n''. This is also called the square-free part of the integer ''n''. Then the square-free ''n'' are exactly the solutions of ''n'' = ''rad''(''n''). DISTRIBUTION OF SQUARE-FREE NUMBERS If ''Q''(''x'') denotes the number of square-free integers between 1 and ''x'', then : (see Pi and Big O Notation ). The asymptotic/ Natural Density of square-free numbers is therefore : where ζ is the Riemann Zeta Function . Likewise, if ''Q''(''x'',''n'') denotes the number of ''n''th power-free integers between 1 and ''x'', one can show : ERDöS SQUAREFREE CONJECTURE The Central Binomial Coefficient is never squarefree for ''n'' > 4. This was proven in 1996 by Olivier Ramaré and Andrew Granville . |
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