Date: 2017-02-15 01:31 From: Hans Havermann <gladhobo@bell.net> To: wclark@mail.usf.edu, math-fun <math-fun@mailman.xmission.com> Reply-To: math-fun <math-fun@mailman.xmission.com> W. Edwin Clark: "It would be interesting to see a reference to a publication that denies that 1 is a factor of every integer." The 19th century arithmetic texts that I had occasion to look at all agreed that both the multiplicand and multiplier of a product were factors and generally gave examples in pairs that excluded the number itself multiplied by one. Edward Liddell in his Arithmetic for the Use of Schools (1860) made it explicit: "We have already learnt that when 6 and 8 are multiplied together they make the product 48. For this reason 6 and 8 are called factors of 48, from the Latin facio, which signifies to make. We may therefore separate or resolve 48 into the two factors 6 and 8, or into the two factors 4 and 12. The figure 1 is not regarded as a factor; consequently, in whole numbers, all factors are greater than unity." Eric Weisstein cites Ore's 1988 Number Theory and its History and Burton's 1989 Elementary Number Theory for his number theoretic usage that "a factor of a number n is equivalent to a divisor of n". He adds: "In elementary education, the term 'factor' is sometimes used to mean proper divisor, i.e., a factor of n other than the number itself. However, as a result of the confusion this practice creates and its inconsistency with the mathematical literature, it should be discouraged." _____________ Gary Snethen (privately) wrote: Bill, Google has 13,900 hit for "prime numbers have two factors" and only 3 hits for "prime numbers have one factor". Two factors: https://www.google.com/webhp?sourceid=chrome-instant&ion=1& espv=2&ie=UTF-8#q=%22prime+numbers+have+two+factors%22 One factor: https://www.google.com/webhp?sourceid=chrome-instant&ion=1& espv=2&ie=UTF-8#q=%22prime+numbers+have+one+factor%22 Dictionaries are rapidly conceding defeat to the vernacular. In a post-truth world, perhaps mathematics will ultimately do the same. ;) On Tue, Feb 14, 2017 at 9:16 AM, Bill Gosper <billgosper@gmail.com> (privately) wrote:
There's a crucial distinction between *factor* and *divisor*! 1 can not be a factor of anything, not even 1! In the multiplicative world, 1 is the empty product--a 0th power--the product of zero things. How many factors of 1 go into 6? 1 is a divisor of everything and a factor of nothing. That's why it's not a prime. --Bill
It looks too late to turn this ship around. But by the reasoning that 1 is a multiplicative factor, then 0 is an additive part, and the function that counts unrestricted partitions needs to be increased everywhere by 1. --rwg