a d̶e̶t̶e̶r̶m̶i̶n̶i̶s̶t̶i̶c̶ test for primes p=1 or p=7 mod 8 with 2 (??) Selfridges

[bhelmes]

Pleasant easter days,

I present a new prime algorithm:

I consider the order in the complex field reduced to the unit circle and count the rational numbers which can be „mapped“ on the unit circle for the different angles.
As every rational point on the unit circle has 8 „combined“ rational points
(tangens = cotagans and mirrored in the four quaters), you can count the rational points.
If the amount of the rational points are equal to the complex order in the unit circle of f,
f is sure to be prime.

This is a discussion paper in the attachment,

if you find an error, regard it as easter egg.

[paulunderwood]

Please give us a worked example for the prime 97.

[bhelmes]

Originally Posted by paulunderwood

Please give us a worked example for the prime 97.

http://devalco.de/unit_circle/system...le.php?prim=97


This seems to be ok.

[Batalov]

What is 'deterministic' about this test?

[Dr Sardonicus]

1. Choose a complex gaussian number a+bi

with the norm a²+b² < f and
with gcd (a,b)=1,
with gcd (b, f)=1 and
jacobi (a²+b², f)=-1 (the norm should be a non quadratic residue)

"Choose" how, exactly?

2. Calculate the exponent

a) f=1 mod 8 exp:=(p-1)/4
b) f=7 mod 8 exp:=(p+1)/4

What is p? It is not previously defined.

Giving up...

[bhelmes]

1. Choose a complex gaussian number a+bi

with the norm a2+b2 < f and
with gcd (a,b)=1,
with gcd (b, f)=1 and
jacobi (a2+b2, f)=-1 (the norm should be a non quadratic residue)

>"Choose" how, exactly?

These are the red elements in the graphic:
http://devalco.de/unit_circle/system_unit_circle.php
(I think this is a wonderful applet, programmed in php)

2. Calculate the exponent

a) f=1 mod 8 exp:=(p-1)/4
b) f=7 mod 8 exp:=(p+1)/4

> What is p? It is not previously defined.

This was a "thinking error" or in german a "Denkfehler"

a) f=1 mod 8 exp:=(f-1)/4
b) f=7 mod 8 exp:=(f+1)/4

[Dr Sardonicus]

Originally Posted by bhelmes

These are the red elements in the graphic:
http://devalco.de/unit_circle/system_unit_circle.php
(I think this is a wonderful applet, programmed in php)

No algorithm is given, and the applet is restricted to odd numbers less than 1000. I can factor odd numbers less than 1000 in my head.

Originally Posted by Dr Sardonicus

What is p? It is not previously defined.

This was a "thinking error" or in german a "Denkfehler"

a) f=1 mod 8 exp:=(f-1)/4
b) f=7 mod 8 exp:=(f+1)/4

[bhelmes]

> No algorithm is given, and the applet is restricted to odd numbers less than 1000. I can factor odd numbers less than 1000 in my head.


I thought that you could use every non quadratic residue with the described conditions, but unfortunetly this is not true:
4+7i is not a good candidate for

http://devalco.de/unit_circle/system...le.php?prim=89.


I called my paper a discussion paper, therefore perhaps someone has some better ideas.


The algorithm should not be a probablistic test for primes,
therefore I thought "deterministic " should be the right term.

[bhelmes]

I think you have to add one condition:
1. b) (a+bi)^(exp/8)=/=1+i mod f

[Batalov]

Let's make a short test that will help us define the runtime of this test (as if it worked, though no data is presented that it does).

Here is the question:

Compare two short processes:
Process A. (aka 3.) Calculate and verify that (a+bi)^exp = 1+i mod f
Process B. Calculate and verify that a^exp = 1 mod f

Is it true that these processes require the same time?
You seem to assume that to be true.

[bhelmes]

A peaceful day for you, Batalov

it would be gentle from you to give a counterexample for the test
before you delete the "deterministic" word in the title.

There was a lot of joy in the mathematical world when the paper "Primes in P" appeared, why not follow this thread ?

Covid-19- time is quite boring but perhaps the situation will become better.

Have a pleasant time,

Bernhard