The Number 1 and the Prime Numbers.

[Great Again]

You already know the question "whether 1 is 0.999... or not" from the meanwhile very well visited thread "Is 1 = 0.999.... ?".

I would now like to confront you with another question: Is it correct that for some time now the number 1 may no longer be considered a prime number?

The definition of a prime number is the following one:

"A prime number is a natural number greater than 1 that is not a product of two smaller natural numbers."

"Greater than one". Should we question this this part of the definition or not? That's the question.

Wikipedia wrote about the problem of the number 1 and the prime numbers:

Primality of one.

Most early Greeks did not even consider 1 to be a number,[35][36] so they could not consider its primality. A few mathematicians from this time also considered the prime numbers to be a subdivision of the odd numbers, so they also did not consider 2 to be prime. However, Euclid and a majority of the other Greek mathematicians considered 2 as prime. The medieval Islamic mathematicians largely followed the Greeks in viewing 1 as not being a number.[35] By the Middle Ages and Renaissance mathematicians began treating 1 as a number, and some of them included it as the first prime number.[37] In the mid-18th century Christian Goldbach listed 1 as prime in his correspondence with Leonhard Euler; however, Euler himself did not consider 1 to be prime.[38] In the 19th century many mathematicians still considered 1 to be prime,[39] and lists of primes that included 1 continued to be published as recently as 1956.[40][41]

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.[39] 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.[41] 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.[42] By the early 20th century, mathematicians began to agree that 1 should not be listed as prime, but rather in its own special category as a "unit".[39]

Source: https://en.wikipedia.org/wiki/Prime_num ... ity_of_one .

Sieve of Eratosthenes (without 1):

Prime_Numbers_Sieve_of_Eratosthenes.gif (153.76 KiB) Viewed 10432 times

Gaussian prime numbers with norm less than_500

Gaussian_Prime_Numbers_With_Norm_Less_Than_500.png (28.09 KiB) Viewed 10426 times

[MagsJ]

_
I’m undecided for now, as 1 is an anomaly in having only one factor.. itself.
but yet the maths would categorise it as a prime number, regardless.

[Destiny]

No because the trick of a prime number is that it isnt 1 but still divisible only by 1. (and itself)

Otherwise its not so special. The fact 1 can only be divided by 1 is the same as that it can only be divided by itself.

Prime numbers are more special. Even though theyre not numero uno.

[obsrvr524]

Primality of one.

Most early Greeks did not even consider 1 to be a number,[35][36] so they could not consider its primality. A few mathematicians from this time also considered the prime numbers to be a subdivision of the odd numbers, so they also did not consider 2 to be prime. However, Euclid and a majority of the other Greek mathematicians considered 2 as prime. The medieval Islamic mathematicians largely followed the Greeks in viewing 1 as not being a number.[35]

Something seems a little off kilter with that narrative. Why would even the Greeks choose to exclude 1 from the set of numbers?

\(\frac{4}{7-3} = what? = nothing?\)

But I guess it is irrelevant now.

By the Middle Ages and Renaissance mathematicians began treating 1 as a number, and some of them included it as the first prime number.[37] In the mid-18th century Christian Goldbach listed 1 as prime in his correspondence with Leonhard Euler; however, Euler himself did not consider 1 to be prime.[38] In the 19th century many mathematicians still considered 1 to be prime,[39] and lists of primes that included 1 continued to be published as recently as 1956.[40][41]

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.[39] 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.[41] 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.[42] By the early 20th century, mathematicians began to agree that 1 should not be listed as prime, but rather in its own special category as a "unit".[39]

And that seems a little too "Minority Report" for my taste - as though someone wanted a perfect score so they just disqualified anything that didn't fit their narrative - much like "We didn't get caught in too much fraud so let's just say the election was absolutely 'perfect' and disclaim any counter evidence".

Or perhaps I am just too politically minded.

I think to give a responsible answer we would have to know the purpose in declaring prime numbers at all. Not being a mathematician, I haven't got a clue. The above analyses don't mention the only issue that would count - why anyone would care. And when that happens, it becomes entirely political very quickly - who stood to lose or gain what? And who or what stands to gain now?

Elections used to have purpose. I suspect everyone forgot that too.

[Great Again]

Primality of one.

Most early Greeks did not even consider 1 to be a number,[35][36] so they could not consider its primality. A few mathematicians from this time also considered the prime numbers to be a subdivision of the odd numbers, so they also did not consider 2 to be prime. However, Euclid and a majority of the other Greek mathematicians considered 2 as prime. The medieval Islamic mathematicians largely followed the Greeks in viewing 1 as not being a number.[35]

Something seems a little off kilter with that narrative. Why would even the Greeks choose to exclude 1 from the set of numbers?

\(\frac{4}{7-3} = what? = nothing?\)

Because it is Wikipedia.

You are right with your criticism. Especially the Ancient Greeks had a strong interest in 1.

But I guess it is irrelevant now.

Yes, of course.

By the Middle Ages and Renaissance mathematicians began treating 1 as a number, and some of them included it as the first prime number.[37] In the mid-18th century Christian Goldbach listed 1 as prime in his correspondence with Leonhard Euler; however, Euler himself did not consider 1 to be prime.[38] In the 19th century many mathematicians still considered 1 to be prime,[39] and lists of primes that included 1 continued to be published as recently as 1956.[40][41]

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.[39] 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.[41] 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.[42] By the early 20th century, mathematicians began to agree that 1 should not be listed as prime, but rather in its own special category as a "unit".[39]

And that seems a little too "Minority Report" for my taste - as though someone wanted a perfect score so they just disqualified anything that didn't fit their narrative - much like "We didn't get caught in too much fraud so let's just say the election was absolutely 'perfect' and disclaim any counter evidence".

Or perhaps I am just too politically minded.

I think to give a responsible answer we would have to know the purpose in declaring prime numbers at all. Not being a mathematician, I haven't got a clue. The above analyses don't mention the only issue that would count - why anyone would care. And when that happens, it becomes entirely political very quickly - who stood to lose or gain what? And who or what stands to gain now?

Elections used to have purpose. I suspect everyone forgot that too.

Yea. Forget Wikipedia anyway. Just note my question in the op:

"Is it correct that for some time now the number 1 may no longer be considered a prime number?"

This is actually the only relevant question when it comes to the number 1 and the problem with the prime numbers concerning the number 1 as well.

[obsrvr524]

ust note my question in the op:

"Is it correct that for some time now the number 1 may no longer be considered a prime number?"

This is actually the only relevant question when it comes to the number 1 and the problem with the prime numbers concerning the number 1 as well.

Without knowing the purpose of declaring prime numbers, I can't answer even whether I care. Can you tell us the significance of such numbers? I honestly have no idea what they are used for - so I can't opine on whether 1 should be included or not.

[MagsJ]

Without knowing the purpose of declaring prime numbers, I can't answer even whether I care. Can you tell us the significance of such numbers?

Primes are used in several routines in information technology, such as public-key cryptography, which relies on the difficulty of factoring large numbers into their prime factors. In abstract algebra, objects that behave in a generalized way like prime numbers include prime elements and prime ideals.

https://en.wikipedia.org › wiki › Prime_number

A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, 1 × 5 or 5 × 1, involve 5 itself. However, 4 is composite because it is a product (2 × 2) in which both numbers are smaller than 4. Primes are central in number theory because of the fundamental theorem of arithmetic: every natural number greater than 1 is either a prime itself or can be factorized as a product of primes that is unique up to their order.

[obsrvr524]

Without knowing the purpose of declaring prime numbers, I can't answer even whether I care. Can you tell us the significance of such numbers?

Primes are used in several routines in information technology, such as public-key cryptography, which relies on the difficulty of factoring large numbers into their prime factors. In abstract algebra, objects that behave in a generalized way like prime numbers include prime elements and prime ideals.

https://en.wikipedia.org › wiki › Prime_number

A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, 1 × 5 or 5 × 1, involve 5 itself. However, 4 is composite because it is a product (2 × 2) in which both numbers are smaller than 4. Primes are central in number theory because of the fundamental theorem of arithmetic: every natural number greater than 1 is either a prime itself or can be factorized as a product of primes that is unique up to their order.

That all sounds like someone was trying to figure out the minimum number of characters required to express all possible quantities. I think the ancient Chinese and Hebrews got into that long ago (even all possible thoughts). And if that is the issue, then I think that certainly 1 should be included.

But today I'm sure encryption is a far far more relevant issue. On the other hand, I'm sure a computer doesn't look in a dictionary to see if a number is called "prime" before it does whatever it is going to do - so why should they care.

So far it seems like -- flip a croc and just see if he lands on his belly.

[MagsJ]

On the other hand, I'm sure a computer doesn't look in a dictionary to see if a number is called "prime" before it does whatever it is going to do..

You seem to be assuming, that computers think independently from their pre-programmed operative/s?

[obsrvr524]

On the other hand, I'm sure a computer doesn't look in a dictionary to see if a number is called "prime" before it does whatever it is going to do..

You seem to be assuming, that computers think independently from their pre-programmed operative/s?

And you seem to be forgetting about missing the joke.

[Great Again]

ust note my question in the op:

"Is it correct that for some time now the number 1 may no longer be considered a prime number?"

This is actually the only relevant question when it comes to the number 1 and the problem with the prime numbers concerning the number 1 as well.

Without knowing the purpose of declaring prime numbers, I can't answer even whether I care. Can you tell us the significance of such numbers? I honestly have no idea what they are used for - so I can't opine on whether 1 should be included or not.

Prime numbers play an important role in information security and especially in the encryption of messages (see cryptography). They are often used in asymmetric cryptosystems such as public-key encryption schemes. Important examples are the Diffie-Hellman key exchange, the RSA cryptosystem used in OpenPGP, among others, the Elgamal cryptosystem and the Rabin cryptosystem. In these, keys are calculated from large, randomly generated prime numbers that must remain secret.

Such algorithms are based on one-way functions that can be executed quickly, but whose inversion is practically impossible to compute with currently known technology. However, new information technologies, for example quantum computers, could change that. The unsolved P-NP problem is related to this. The P-NP problem (also P≟NP, P versus NP) is an unsolved problem in mathematics, specifically complexity theory in theoretical computer science. The question here is whether the set of all problems that can be solved quickly (P {\displaystyle P} P) and the set of all problems for which one can quickly check a proposed solution for correctness (N P {\displaystyle NP} NP) are identical.

Some species of animals and plants (e.g., certain cicadas or spruce trees) reproduce especially strongly in cycles of primes (say, every 11, 13, or 17 years) to make it difficult for predators to adjust to the mass occurrence.

What I am concerned with in this topic is the answering of the question and the following straight argumentation, whether and why the 1 should be counted again to the prime numbers or not. So basically I am concerned with logic and with straight arguments. It is - as I already said - similar to the question whether 1 and 0.999... are equal or not (I say: they are not).

The more ridiculous than serious argument that "1 is not allowed to be a prime number, because 1 is divisible only by itself and 1" actually means that 1 is a prime number, because the definition that "a prime number is divisible only by itself and 1" also applies to 1.

But then it is said that every prime number has two different divisors, but the number 1 has only itself. Yes, but the original definition does not say anything about the separability of the divisors of a prime number, but only that a prime number is divisible only by itself and by 1.

The original definition has been changed so that one can claim afterwards that another definition than the valid one is "too complicated". In reality, it's the other way around.

[obsrvr524]

The original definition has been changed so that one can claim afterwards that another definition than the valid one is "too complicated". In reality, it's the other way around.

That just sounds like a sign of our times. They do the same with movies and political narratives - keep the name but change the characters and narrative so as to hypnotically rewrite history and people's beliefs and behaviors (which irritates me to a degree - but I'll get over it).

And hate to harp on this but - it seems there are two concerns being addressed -

• What should be done
• What is logically correct

The first depends on usefulness - and I don't see anyone on this board as enough of a mathematician to contribute much of a meaningful response.

And the second is entirely an issue of the declared definition. I dislike people changing definitions (or words) just to manipulate other concerns, but since that one happened long ago - there is little point in arguing about it now. If you were to ask if it is logically correct that someone called a "democrat" supports socialism - I would have to say - "of course not - those are opposites". But again - just the times we suffer through.

Going back to the original definition of "prime number" (where 1 was not excluded) obviously 1 would still logically qualify (whether mathematicians liked it or not). But living in the present after 1 has been disqualified by newspeak definition of "prime number" - it is obvious that logically 1 cannot be called "prime" any longer. It is like democracy - once it's gone - it's gone for good.

Other than some other usefulness issue, I don't see what else can be said - we live in a time of adopting newspeak so as to accomplish purposes that others dictate. And after any definition has been altered (for whatever reason), the logic must adopt that new definition.

And again - I don't see how computers would care since they will do whatever they are told regardless of anyone's defined terminology.

[MagsJ]

On the other hand, I'm sure a computer doesn't look in a dictionary to see if a number is called "prime" before it does whatever it is going to do..

You seem to be assuming, that computers think independently from their pre-programmed operative/s?

And you seem to be forgetting about missing the joke.

1 should be called something else, because it hasn’t got the properties of primes.. of which it lacks the encryptological integrity of.

[obsrvr524]

1 should be called something else, because it hasn’t got the properties of primes.. of which it lacks the encryptological integrity of.

Perhaps call the other primes - "crypto-primes".

[Great Again]

The original definition was: "A prime number is a natural number that is divisible exclusively by itself and 1".

But now the definition is: "A prime number is a natural number that is greater than 1 and divisible exclusively by itself and 1".

That is cheating!

[MagsJ]

1 should be called something else, because it hasn’t got the properties of primes.. of which it lacks the encryptological integrity of.

Perhaps call the other primes - "crypto-primes".

Perhaps

No rush

[MagsJ]

The original definition was: "A prime number is a natural number that is divisible exclusively by itself and 1".

But now the definition is: "A prime number is a natural number that is greater than 1 and divisible exclusively by itself and 1".

That is cheating!

So what do you think that means or is inferring to?

[obsrvr524]

That is cheating!

Isn't that the prevailing morality? - "Thy rules are thus - but not for us."

[Kathrina]

The original definition was: "A prime number is a natural number that is divisible exclusively by itself and 1".

But now the definition is: "A prime number is a natural number that is greater than 1 and divisible exclusively by itself and 1".

That is cheating!

So what do you think that means or is inferring to?

If I may answer:

It more or less reflects what we have to deal with more and more in modern times: the rulers enforce everything in the way that suits them best, and this is often to the detriment of all other people. The sciences are becoming more and more dependent, the culture as a whole is becoming more and more victimized.

[Great Again]

The original definition was: "A prime number is a natural number that is divisible exclusively by itself and 1".

But now the definition is: "A prime number is a natural number that is greater than 1 and divisible exclusively by itself and 1".

That is cheating!

So what do you think that means or is inferring to?

If I may answer:

It more or less reflects what we have to deal with more and more in modern times: the rulers enforce everything in the way that suits them best, and this is often to the detriment of all other people. The sciences are becoming more and more dependent, the culture as a whole is becoming more and more victimized.

Agreed.

That is cheating!

Isn't that the prevailing morality? - "Thy rules are thus - but not for us."

Absolutely.

[MagsJ]

_
Indeed, historically many mathematicians up to the nineteenth century thought of 1 as prime – Henri Lebesgue (1875–1941) is usually said to have been the last professional mathematician to call 1 prime. (The Greeks didn't regard 1 as prime, but that's because they didn't regard it as a number at all!)

http://www.foster77.co.uk/Foster,%20Mathematics%20in%20School,%20Why%20isn%27t%201%20a%20prime%20number.pdf
Why isn't 1 a prime number? - Colin Foster
_
https://blogs.scientificamerican.com/roots-of-unity/why-isnt-1-a-prime-number/
Why Isn't 1 a Prime Number? - Scientific American Blog Network
2 Apr 2019 — For that reason, 1 couldn't have been prime — it wasn't even a number. Ninth-century Arab mathematician al-Kindī wrote that it was not a number and therefore not even or odd. The view that 1 was the building block for all numbers but not a number itself lasted for centuries.
_
2 is prime. Rebuttal: Because even numbers are composite, 2 is not a prime.

___
- 1 wasn’t considered a number by the Greeks and Arabs? Who knew..
- Why can’t 2 be both prime and composite?

Does the OECD Math Department know about this? I wonder what their take is on such numerical anomalies?
As philosophers, I say we have free-rein on such matters, and declare them as how we see fit.

..and so ..I’m relabelling 1, and declaring 2 of dual numerical status.

[obsrvr524]

What I am curious about is why 1 wasn't considered a "number". That just seems bizarre.

"Four people voted - list the number of votes" -

• 3 votes in favor
  
• none against.

Maybe it was one of those "perfect fraud-free election" things.

[Great Again]

_
Indeed, historically many mathematicians up to the nineteenth century thought of 1 as prime – Henri Lebesgue (1875–1941) is usually said to have been the last professional mathematician to call 1 prime. (The Greeks didn't regard 1 as prime, but that's because they didn't regard it as a number at all!)

http://www.foster77.co.uk/Foster,%20Mathematics%20in%20School,%20Why%20isn%27t%201%20a%20prime%20number.pdf
Why isn't 1 a prime number? - Colin Foster
_
https://blogs.scientificamerican.com/roots-of-unity/why-isnt-1-a-prime-number/
Why Isn't 1 a Prime Number? - Scientific American Blog Network
2 Apr 2019 — For that reason, 1 couldn't have been prime — it wasn't even a number. Ninth-century Arab mathematician al-Kindī wrote that it was not a number and therefore not even or odd. The view that 1 was the building block for all numbers but not a number itself lasted for centuries.
_
2 is prime. Rebuttal: Because even numbers are composite, 2 is not a prime.

___
- 1 wasn’t considered a number by the Greeks and Arabs? Who knew..
- Why can’t 2 be both prime and composite?

Does the OECD Math Department know about this? I wonder what their take is on such numerical anomalies?
As philosophers, I say we have free-rein on such matters, and declare them as how we see fit.

..and so ..I’m relabelling 1, and declaring 2 of dual numerical status.

Yes, the number 2 is also a problem. Either we allow the number 1 to be a prime number again, or we also take the number 2 out of the prime number set and say: "Within the whole positive number set, the number 2 can only be divided by itself and by 1, because there is only one positiv number which is smaller than 2". So then the new definition would be: "A prime number is that integer positive number which is greater than 2 and divisible only by itself and 1". In this case, the prime numbers would begin with the number 3.

This would lead to other problems. So we should leave it at the first definition, so that 1 and 2 are prime numbers: "A prime number is that integer positive number which is divisible by itself and 1".

[MagsJ]

What I am curious about is why 1 wasn't considered a "number". That just seems bizarre.

1 was considered a unit, and a number was composed of multiple units, is why.

"Four people voted - list the number of votes" -

• 3 votes in favor
  
• none against.

Maybe it was one of those "perfect fraud-free election" things.

Lol


Should the number 1 still not be allowed to be a prime number?
You may select 1 option

Yes. 3 43%

No. 4. 57%

I do not care at all. 0. No votes

Total votes : 7

[MagsJ]

Yes, the number 2 is also a problem. Either we allow the number 1 to be a prime number again, or we also take the number 2 out of the prime number set and say: "Within the whole positive number set, the number 2 can only be divided by itself and by 1, because there is only one positiv number which is smaller than 2". So then the new definition would be: "A prime number is that integer positive number which is greater than 2 and divisible only by itself and 1". In this case, the prime numbers would begin with the number 3.

This would lead to other problems. So we should leave it at the first definition, so that 1 and 2 are prime numbers: "A prime number is that integer positive number which is divisible by itself and 1".

I guess the Mathematician should know what numbers are applicable to use in what calculations and circumstance, regardless of category.