Integers without representation
Let N(a1, . . . , an) be the number of integers without nonnegative integer representations by a1, . . . , an. In this chapter, a thorough presentation of the function N(a1, . . . , an) is given. In 1882, Sylvester, obtained the exact value when n = 2. Then, in 1884 in Educational Times, Sylvester posed (as a recreational problem) the question of finding such a formula. An ingenious solution was given by W. J. Curran Sharp. It remains a mystery why the standard reference to this celebrated formula of Sylvester is the solution given by Curran Sharp rather than its original appearance. In this chapter, the original page of this famous and much cited manuscript is reproduced. It also gives two other proofs of equality (2) and discusses the work of M. Nijenhuis and H. S. Wilf connecting N(a1, . . . , an) to FP as well as to other concepts (such as the Gorenstein condition). Also, some general bounds on N(a1, . . . , an) and exact formulas for special sequences are discussed. A generalization of Sylvester's formula due to Ø. J. Rødseth, where the so-called Bernoulli numbers appeared, is treated. The final section of the chapter is devoted to two ‘integer representation’ games: the well-known Sylver Coinage invented by J. C. Conway, and the jugs problem, which roots can be traced back at least as far as Tartaglia, an Italian mathematician of the 16th century.
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