#perfectsquares

Mersenne Prime Squares —————-April-Mersenne

digital, #4-10, 2021, Reginald Brooks

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LINKS:

http://www.brooksdesign-ps.net/Reginald_Brooks/Code/Html/MSST/TPISC/TPISC_V/MPS.html

http://www.brooksdesign-ps.net/Reginald_Brooks/Code/Html/porfoli5.htm

https://en.wikipedia.org/wiki/Mersenne_prime

https://en.wikipedia.org/wiki/Perfect_number

Mersenne Prime Squares ——————April-Mersenne

digital, #4-11, 2021, Reginald Brooks

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LINKS:

http://www.brooksdesign-ps.net/Reginald_Brooks/Code/Html/MSST/TPISC/TPISC_V/MPS.html

http://www.brooksdesign-ps.net/Reginald_Brooks/Code/Html/porfoli5.htm

https://en.wikipedia.org/wiki/Mersenne_prime

https://en.wikipedia.org/wiki/Perfect_number

April-Mersenne

Mersenne Prime Squares

digital, #4-7, 2021, Reginald Brooks

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LINKS:

http://www.brooksdesign-ps.net/Reginald_Brooks/Code/Html/porfoli5.htm

https://en.wikipedia.org/wiki/Mersenne_prime

https://en.wikipedia.org/wiki/Perfect_number

Mersenne Prime Square (MPS)

April-Mersenne

A Mersenne Prime Square:

Mersenne Prime-Perfect Number

And two Perfect Squares

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What was once a dot

• Mersenne Prime

has become a line

________ Mersenne Prime

and lines become areas

▢ Mersenne Prime Square

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Mersenne Prime Squares: the basics

ABSTRACT:

An ODD x ODD gives an ODD.

Divide such ODD gives an EVEN and ODD (with EVEN>ODD).

Divide that EVEN (≥2) gives EVENS.

Divide that ODD (≥2) gives an EVEN and ODD.

UNIVERSAL:

A z x z gives an z².

Divide such z² gives a xz and yz (with x>y).

Divide that xz (≥2) gives a x² and xy.

Divide that yz (≥2) gives a xy and y².

SPECIFIC:

A Mp x Mp gives a Mp².

Divide such Mp² gives a PN and OC (with PN>OC).

Divide that PN (≥2) gives a PNS and CR.

Divide that OC (≥2) gives a CR and OCS.

EXAMPLE: let z=Mp=7

An 7 x 7 gives a 7² = 49.

Divide such 49 gives a 28=4x7 and 21=3x7 (with 4>3).

Divide that 28 (≥2) gives a 16=4² and 12=3x4.

Divide that 21 (≥2) gives a 12=3x4 and 9=3².

KEY: Mp = Mersenne Prime, Mp² = MPS = Mersenne Prime Square, PN = Perfect Number, PNS = Perfect Number Square, OC = ODD Complement, OCS = ODD Complement Square, CR = Complement Rectangle. NOTE: ALL NUMBERS = natural whole integer numbers (WIN).

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digital, #4-5, 2021, Reginald Brooks

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LINKS:

http://www.brooksdesign-ps.net/Reginald_Brooks/Code/Html/porfoli5.htm

https://en.wikipedia.org/wiki/Mersenne_prime

https://en.wikipedia.org/wiki/Perfect_number

April-Mersenne

A Mersenne Prime Square:

Mersenne Prime-Perfect Number

And two Perfect Squares

~~~

What was once a dot

• Mersenne Prime

has become a line

________ Mersenne Prime

and lines become areas

▢ Mersenne Prime Square

…the story follows.

~~~

Mersenne Prime Squares: the basics

ABSTRACT:

An ODD x ODD gives an ODD.

Divide such ODD gives an EVEN and ODD (with EVEN>ODD).

Divide that EVEN (≥2) gives EVENS.

Divide that ODD (≥2) gives an EVEN and ODD.

UNIVERSAL:

A z x z gives an z².

Divide such z² gives a xz and yz (with x>y).

Divide that xz (≥2) gives a x² and xy.

Divide that yz (≥2) gives a xy and y².

SPECIFIC:

A Mp x Mp gives a Mp².

Divide such Mp² gives a PN and OC (with PN>OC).

Divide that PN (≥2) gives a PNS and CR.

Divide that OC (≥2) gives a CR and OCS.

EXAMPLE: let z=Mp=7

An 7 x 7 gives a 7² = 49.

Divide such 49 gives a 28=4x7 and 21=3x7 (with 4>3).

Divide that 28 (≥2) gives a 16=4² and 12=3x4.

Divide that 21 (≥2) gives a 12=3x4 and 9=3².

KEY: Mp = Mersenne Prime, Mp² = MPS = Mersenne Prime Square, PN = Perfect Number, PNS = Perfect Number Square, OC = ODD Complement, OCS = ODD Complement Square, CR = Complement Rectangle. NOTE: ALL NUMBERS = natural whole integer numbers (WIN).

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INTRODUCTION

From Euclid’s proof of Perfect Numbers to Euler’s connecting them to Mersenne Primes, the universal and ongoing mystery remains to this day. Why is there this connection? What is it trying to tell us about the fundamental relationship between EVEN (all known Perfect Numbers are EVEN) and ODD (all primes >2 are ODD) numbers — at least in regards to the Mersenne Prime - Perfect Number pairing? Is there a new approach? Yes!

A brief reference:

Mersenne Primes

In mathematics, a Mersenne prime is a prime number that is one less than a power of two.That is, it is a prime number of the form Mn =2ⁿ −1 for some integer n. They are named after Marin Mersenne, a French Minim friar, who studied them in the early 17th century. If n is a composite number then so is 2ⁿ− 1. Therefore, an equivalent definition of the Mersenne primes is that they are the prime numbers of the form Mp = 2ᵖ -1 for some prime p.

Reference: Wikipedia: Mersenne Prime

Perfect Numbers

Mersenne primes Mp are closely connected to perfect numbers. In the 4th century BC, Euclid proved that if 2ᵖ -1 is prime, then 2ᵖ⁻¹( 2ᵖ -1) is a perfect number. In the 18th century, Leonhard Euler proved that, conversely, all even perfect numbers have this form.[4] This is known as the Euclid–Euler theorem. It is unknown whether there are any odd perfect numbers.

Reference: Wikipedia: Mersenne Prime

In number theory, a perfect number is a positive integer that is equal to the sum of its positive divisors, excluding the number itself. For instance, 6 has divisors 1, 2 and 3 (excluding itself), and 1 + 2 + 3 = 6, so 6 is a perfect number.

Reference: Wikipedia: Perfect number

Euclid–Euler theorem

The Euclid–Euler theorem is a theorem in mathematics that relates perfect numbers to Mersenne primes. It states that an even number is perfect if and only if it has the form 2ᵖ⁻¹(2ᵖ -1), where 2ᵖ -1 is a prime number. The theorem is named after Euclid and Leonhard Euler.

It has been conjectured that there are infinitely many Mersenne primes. Although the truth of this conjecture remains unknown, it is equivalent, by the Euclid–Euler theorem, to the conjecture that there are infinitely many even perfect numbers. However, it is also unknown whether there exists even a single odd perfect number.[1]

Reference: Wikipedia: Euclid-Euler theorem

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digital, #4-3, 2021, Reginald Brooks

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…the story follows.

LINKS:

http://www.brooksdesign-ps.net/

https://en.wikipedia.org/wiki/Mersenne_prime

https://en.wikipedia.org/wiki/Perfect_number

Mersenne Prime Squares ——————-April-Mersenne

digital, #4-13, 2021, Reginald Brooks

~~~

LINKS:

http://www.brooksdesign-ps.net/Reginald_Brooks/Code/Html/MSST/TPISC/TPISC_V/MPS.html

http://www.brooksdesign-ps.net/Reginald_Brooks/Code/Html/porfoli5.htm

https://en.wikipedia.org/wiki/Mersenne_prime

https://en.wikipedia.org/wiki/Perfect_number

Mersenne Prime Squares ————-April-Mersenne

digital, #4-26, 2021, Reginald Brooks

~~~

LINKS:

http://brooksdesign-ps.net/reginald_brooks/code/html/netarti5.htm

http://www.brooksdesign-ps.net/Reginald_Brooks/Code/Html/porfoli5.htm

https://en.wikipedia.org/wiki/Mersenne_prime

https://en.wikipedia.org/wiki/Perfect_number

Mersenne Prime Squares ————-Apritl-Mersenne

digital, #4-18, 2021, Reginald Brooks

There is natural connection between the previously covered Goldbach Conjecture (all Evens composed of the sums of two primes) and, now, the geometry of the Mersenne Prime Square!

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LINKS:

http://brooksdesign-ps.net/reginald_brooks/code/html/netarti5.htm

http://www.brooksdesign-ps.net/Reginald_Brooks/Code/Html/porfoli5.htm

https://en.wikipedia.org/wiki/Mersenne_prime

https://en.wikipedia.org/wiki/Perfect_number