Prime Numbers Book

paulunderwood · 2020-10-10, 13:40

Enjoy and criticize. It ain't no C&P but might help some.

After 20 downloads, I have made the changes to typos and Mersenne data. A new copy is now uploaded.

2 More downloads and I have uploaded a new version without the Euler Phi Function nonsense -- Thanks Robert.

24 more downloads and a new version is up with accumulated corrections and suggested text.

13 more downloads. The replacement (12th Oct) version corrects some grammatical errors, includes suggestions by Nick and a Lucas Sequence algorithm.

12 more downloads. The replacement (Oct 15th) makes the suggestions by Nick and corrects some typos.

4 more downloads. The replacement builds on suggestions by Nick. There is now a section on Gaussian primes. Replacement (18 Oct,2020) uploaded.

Dr Sardonicus · 2020-10-10, 14:10

There is a caveat in using the term "number" to mean "natural number" or "positive integer."

Euclid distinguished between 1 ("the unit") and integers greater than 1, which he called "numbers." Thus, a "prime number" was, by definition an integer greater than 1.

The unit, 1, is neither prime nor composite.

kriesel · 2020-10-10, 14:28

Legendre, Jocobi and Kronecker (should be Jacobi) (page 3)

"The determinant of a A is" (bottom of page 9)
The determinant of a matrix A is

nothing else conspicuous through page 10 on a first pass.

Pretty clean.

Viliam Furik · 2020-10-10, 14:52

Page 3 - Typo in "Jocobi"
Page 32 - Fig. 7.1 contains n = 2 twice, lots of unknown where it shouldn't be (n= 61,89,107)

R. Gerbicz · 2020-10-10, 15:59

On page 39:
"The Euler Phi Function is useful since for all a
a^(1+φ(n)) ≡ a (mod n)."

This is very false, a small counter-example is a=2, n=4.

bhelmes · 2020-10-10, 20:46

"It is important to analize the the exponentiating" page 49.

Greetings
Bernhard

Knut · 2020-10-10, 20:53

You write i were the square root of -1.

As far as I know, the square root r of a number x is the non-negative solution of the equation r^2 = x as far as real numbers are concerned.

As i is neither positive nor negative and i^2 = -1 and (-i)^2 = -1 in my opinion it is not correct to say i would be the square root of -1.

Best regards

Knut

paulunderwood · 2020-10-10, 22:40

Quote:

Originally Posted by Knut

You write i were the square root of -1.

As far as I know, the square root r of a number x is the non-negative solution of the equation r^2 = x as far as real numbers are concerned.

As i is neither positive nor negative and i^2 = -1 and (-i)^2 = -1 in my opinion it is not correct to say i would be the square root of -1.

Best regards

Knut

I have read https://en.wikipedia.org/wiki/Imaginary_unit What would you say I should write? i is fixed to be one of the two square roots of -1?

paulunderwood · 2020-10-10, 22:43

I have tidied up what I want write about "The Fermatian Child".

I will refrain from reposting the paper until more input from folks.

Knut · 2020-10-11, 05:28

I think there is no square root of -1.

Instead, define the "imaginary unit" i with i^2 = -1. And then define complex numbers.

This, in my opinion, could help to avoid disturbances.

E.g.

x^2 = 3*i
x^2 = - 3*i
x^2 = 2 + 3*i
x^2 = 2 - 3*i
x^2 = - 2 + 3*i
x^2 = - 2 - 3*i

There are two (complex) solutions of each equation, of course. Which of the two solutions is to be named the "square root"? I am not aware of a definition.

I apologize fpr my poor command of the English language and hope you can see my point.

Best regards

Knut

Nick · 2020-10-11, 07:44

There are a number of errors at the start, some just details of definitions but others hiding more important points.
How it would be best to resolve these depends strongly on who you are writing this for.
What is the intended audience?

2020-10-10, 14:28	#3
kriesel "TF79LL86GIMPS96gpu17" Mar 2017 US midwest 29×167 Posts	spell check; grammar Legendre, Jocobi and Kronecker (should be Jacobi) (page 3) "The determinant of a A is" (bottom of page 9) The determinant of a matrix A is nothing else conspicuous through page 10 on a first pass. Pretty clean. Last fiddled with by kriesel on 2020-10-10 at 14:29

2020-10-10, 22:43	#9
paulunderwood Sep 2002 Database er0rr 110111001011₂ Posts	I have tidied up what I want write about "The Fermatian Child". I will refrain from reposting the paper until more input from folks. Last fiddled with by paulunderwood on 2020-10-10 at 22:51

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2020-10-10, 14:10	#2
Dr Sardonicus Feb 2017 Nowhere 23·181 Posts	There is a caveat in using the term "number" to mean "natural number" or "positive integer." Euclid distinguished between 1 ("the unit") and integers greater than 1, which he called "numbers." Thus, a "prime number" was, by definition an integer greater than 1. The unit, 1, is neither prime nor composite.

2020-10-10, 14:52	#4
Viliam Furik "Viliam Furík" Jul 2018 Martin, Slovakia 2³·41 Posts	Page 3 - Typo in "Jocobi" Page 32 - Fig. 7.1 contains n = 2 twice, lots of unknown where it shouldn't be (n= 61,89,107)

2020-10-10, 15:59	#5
R. Gerbicz "Robert Gerbicz" Oct 2005 Hungary 1,429 Posts	On page 39: "The Euler Phi Function is useful since for all a a^(1+φ(n)) ≡ a (mod n)." This is very false, a small counter-example is a=2, n=4.

2020-10-10, 20:46	#6
bhelmes Mar 2016 2³×37 Posts	"It is important to analize the the exponentiating" page 49. Greetings Bernhard

2020-10-10, 20:53	#7
Knut Oct 2020 2 Posts	You write i were the square root of -1. As far as I know, the square root r of a number x is the non-negative solution of the equation r^2 = x as far as real numbers are concerned. As i is neither positive nor negative and i^2 = -1 and (-i)^2 = -1 in my opinion it is not correct to say i would be the square root of -1. Best regards Knut

2020-10-11, 05:28	#10
Knut Oct 2020 2 Posts	I think there is no square root of -1. Instead, define the "imaginary unit" i with i^2 = -1. And then define complex numbers. This, in my opinion, could help to avoid disturbances. E.g. x^2 = 3i x^2 = - 3i x^2 = 2 + 3i x^2 = 2 - 3i x^2 = - 2 + 3i x^2 = - 2 - 3i There are two (complex) solutions of each equation, of course. Which of the two solutions is to be named the "square root"? I am not aware of a definition. I apologize fpr my poor command of the English language and hope you can see my point. Best regards Knut

2020-10-11, 07:44	#11
Nick Dec 2012 The Netherlands 2·7·113 Posts	There are a number of errors at the start, some just details of definitions but others hiding more important points. How it would be best to resolve these depends strongly on who you are writing this for. What is the intended audience?