## 1 Is Not Prime

Arithmetic, algebra, number theory, sequence and series, analysis, ...
kenbrooker
Posts: 137
Joined: Mon Feb 19, 2018 3:05 am
Location: Oregon, USA

### 1 Is Not Prime

By euler
This thread has been created to provide a place to discuss the primality of 1. The posts here were originally found in the Clarification thread for problem 618.
traxex wrote:
Wed Jan 17, 2018 7:58 pm
The number 1 is not a prime. The only prime factor of 5 is itself, which is true for all primes.

Otherwise you would have 5 = 1*5 = 1*1*5 = ..., so the prime factorization would not be unique, which would complicate many definitions.

In the past 1 was often considered a prime. See Wikipedia for a bit of interesting history. Seems like Euler was ahead of his time in this as well.
LarryBlake wrote:
Sat Jan 20, 2018 1:46 pm
Another good link is https://en.wikipedia.org/wiki/Fundament ... arithmetic .
[/edit]
traxex wrote:
Wed Jan 17, 2018 7:58 pm
The number 1 is not a prime. The only prime factor of 5 is itself, which is true for all primes.

Otherwise you would have 5 = 1*5 = 1*1*5 = ..., so the prime factorization would not be unique, which would complicate many definitions.

In the past 1 was often considered a prime. See Wikipedia for a bit of interesting history. Seems like Euler was ahead of his time in this as well.
traxex... I agree, the history of whether or not 1 is prime is both "interesting" and, to me, discouragingly less than rigorous, as follows, in Wikipedia --

"If the definition of a prime number were changed to call 1 a prime, many statements involving prime numbers would need to be reworded in a more awkward way. For example, the fundamental theorem of arithmetic would need to be rephrased in terms of factorizations into primes greater than 1, because every number would have multiple factorizations with different numbers of copies of 1. Similarly, the sieve of Eratosthenes would not work correctly if it handled 1 as a prime, because it would eliminate all multiples of 1 (that is, all other numbers) and output only the single number 1. Some other more technical properties of prime numbers also do not hold for the number 1: for instance, the formulas for Euler's totient function or for the sum of divisors function are different for prime numbers than they are for 1." [Source: Wikipedia]

Please consider: Since when should formal logical decisions be based upon (completely) avoiding rewording? Or, based upon perpetuating (the consistency of) sieves? By the same arbitrary and capricious "standards" we might easily still maintain that the sun revolves around planet Earth... We could also eliminate zero as a "number" because one factorial equals one and all (other) factorials are unique, and zero factorial equaling one is not "unique"! And re "technical properties," of sums of quotients, while one-third plus one-third plus one-third is agreed to equal one, 0.333-> plus 0.333-> plus 0.333-> must also be agreed to equal 1.0 (for "consistency") even though 0.999-> clearly and explicitly, even to a ("complete") layman, does not equal one... Just try continually stepping nine-tenths of the way to a wall, starting one meter away... You will never reach the wall...

What I see in the Wikipedia history of whether or not 1 is prime (today : ) overlooks the first and foremost acknowledgement that Logic is a subset of Philosophy (the root of "PhD"s) and Mathematics is a subset of Logic; and, the Eulerian-equal Logician -- Kurt Gödel proved, using prime numbers in an ingenious way, that any non-trivial system of Mathematics is either "Complete" or "Consistent" but never both. This, to my admittedly low-level interpretation of Gödel's two Incompleteness Theorems, left Russell and Whitehead with no rigorous reason to continue to try to write the fantasized Be-All and End-All -- Principia Mathematica -- and leaves Mathematicians, still, with "standards" like completely avoiding rewording and, consistently perpetuating Eratosthenes! Well, that's One interesting and...
pedestrian level of Completeness and Consistency...

ps - Lest anyone think I am "down" on Mathematics,
my second All-Time Hero is...
Ramanujan...
Last edited by kenbrooker on Sun Mar 25, 2018 3:09 am, edited 5 times in total.
"Good Judgment comes from Experience; 