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在讨论质数与合数时,“人教版”小学数学教师用书五年级下册中有这么一句话:“由于自然数是无限的,所以质数和合数也是无限的。”这里的结论——质数和合数是无限的——是正确的,但推理是错误的,或者说理由是不充分的。结论中关于合数的无限性是显然的:大于2的偶数都是合数,而大于2的偶数是无限的,从而合数是无限的。于是关键就在于质数(素数)的个数问题,这是数论中的基本问题。一、为什么不能从自然数的无限性推出质数与合
When discussing the prime numbers and the complex numbers, there is a sentence in the fifth grade of the “PEP” primary school mathematics teacher’s book: “Since the natural numbers are infinite, the prime numbers and the composite numbers are also infinite.” The conclusion here - the prime numbers and the composite numbers are infinite - is correct, but the reasoning is wrong, or the reasons are not sufficient. The conclusion about the infiniteness of the composite number is obvious: even numbers greater than 2 are composite numbers, and even numbers greater than two are infinite, so the composite number is infinite. The key is then the number of primes (prime numbers), which is the basic problem in number theory. First, why can not be introduced from the infinite nature of the prime numbers and together