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March 7

Pros and cons of including 1 as a prime power

Pros:

Every divisor of a prime power would be a prime power, including 1.
The number of divisors of p^k is k + 1, even if k = 0.
The sum of the divisors of p^k is (p^{k + 1} - 1)/(p - 1), even if k = 0.

Cons:

The prime p in n = p^k would not be unique for n = 1 (p could be any prime while k = 0).
The totient formula φ(p^k) = p^k - p^{k - 1} would not be correct for k = 0.
There is no actual field with one element (though there would still be a "fake" field with one element).

So, do most mathematicians consider 1 to be a prime power or not?

GeoffreyT2000 (talk) 04:14, 7 March 2023 (UTC)[reply]

The answer is, no, they don't. A prime power is a number that can be written in the form p^k in which p is a prime number and k is a positive integer. The first prime number is 2 and the first positive integer is 1, so the first element of the sequence of prime powers is equal to 2¹. Historically, 1 has sometimes been considered prime (see Prime number § Primality of one), but the cons against inclusion were found to outweigh the pros. --Lambiam 10:20, 7 March 2023 (UTC)[reply]

Think of 1 as a prime like Pluto as a planet. In both cases, it came down to people deciding what worked best as a definition - in their collective opinion - and letting the cards lie where they fell. Contrary to what some say, the decision did not mean that Pluto was "downgraded", because there's nothing inherently "high" about planets as compared with other celestial objects. Same with 1 as a prime: it fits some very simple definitions of "prime number", but not the one that is used today. 1 hasn't been downgraded; in a sense it's been upgraded, into a category of "units", consisting uniquely of itself. -- Jack of Oz ^{[pleasantries]} 16:28, 8 March 2023 (UTC)[reply]

But 1 was a bit lonely as the only unit among the natural numbers. Fortunately, The Nine Chapters on the Mathematical Art discovered the negative numbers, and thereby the ring of integers, and in this ring 1 got a companion, −1. --Lambiam 18:46, 8 March 2023 (UTC)[reply]

Usually, synergy works its magic spell so that 1 + 1 = more than the sum of the parts. But when 1 is added to -1, we get nothing, annihilation. So they can never have babies, and should consider adoption. -- Jack of Oz ^{[pleasantries]} 21:47, 8 March 2023 (UTC)[reply]

"Go forth and add"? I thought another algebraic operation was involved. --Lambiam 23:37, 8 March 2023 (UTC)[reply]

For all except the adder of course. ;) NadVolum (talk) 23:39, 8 March 2023 (UTC)[reply]

"Flaming 'eck!" This excursion reminds me of when the Talking Heads released their Speaking in Tongues album in '83 and David Byrne sang "Ah, watch out you might get what you're after. Cool babies strange but not a stranger."[1] Adam and Eve were also rebellious (and naked) and an adder (rebracketed from "a nadder" see [2]) could not resist enlightening them. Try -(-1)+1 and -1+(-1). Thus when the newfangled negative numbers were introduced to their natural counterparts within high-society properly clothed, they have since had lots of babies. :-) Naturally, the rounded brackets seem to make the two units appear to be consuming apples. Modocc (talk) 19:17, 10 March 2023 (UTC)[reply]