By Serge Lang

This 5th version of Lang's ebook covers all of the issues ordinarily taught within the first-year calculus series. Divided into 5 elements, each one component of a primary direction IN CALCULUS comprises examples and purposes on the subject of the subject lined. furthermore, the rear of the ebook comprises special strategies to a number of the routines, permitting them to be used as worked-out examples -- one of many major advancements over past variants.

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But since i − j < pk , the larger prime can not divide it, so it must divide the product p1 p2 · · · pk−1 , but these are all smaller primes, another contradiction. Therefore, a prime larger than pk can divide at most one of these numbers. b. Since there are n−k +1 primes from pk up to pn , and each one can divide at most one of the pk numbers p1 p2 · · · pk−1 i − 1, there must be at least one of the numbers which is not divisible by any prime from pk up to pn . ) From part (a), the primes less than pk also do not divide any of the the numbers, in particular, the one whose existence we have just shown.

Thus, d = −1 + b − c. Now, we see that c is either a − d or 10 + d − a. If c = a − d, then d = −1 + b − c = −1 + b − a + d. From this, we arrive at a + 1 = b, a contradiction. Thus c = 10 + d − 1. If a = a − d, then d = 0. Proceeding along with thought, b = c + 1 = 9 + c − b now, which tells us that b = 8, c = 7 and a = 4. This is a contradiction. Thus a = 9 + c − b and b = a − d. We now have four equations in four unknowns. Solving this system, we find that a = 7, b = 6, c = 4, and d = 1. This gives a fixed point, namely 6174.

B. The smallest prime between 5 and 10 is 7. c. The smallest prime between 19 and 38 is 23. d. The smallest prime between 31 and 62 is 37. 7. a. The smallest prime between 4 and 8 is 5. b. The smallest prime between 6 and 12 is 7. c. The smallest prime between 23 and 46 is 29. d. The smallest prime between 47 and 94 is 53. To see that the primes are indeed in the range, we print them as triples (n2 , smallest prime, (n + 1)2 ). For n = 1, 2, . . , 10 we have (1, 2, 4), (4, 5, 9), (9, 11, 16), (16, 17, 25), (25, 29, 36), (36, 37, 49), (49, 53, 64), (64, 67, 81), (81, 83, 100), and (100, 101, 121).

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