A new method of factorization of very large numbers

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A new method of factorization of very large numbers

Jamel Ghanouchi

To cite this version:

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A new method of factorization of very large numbers

Jamel Ghanouchi, ESSTT

Abstract:

This paper presents a new method of factorization of a number, even if it is very large. Keywords: Large number ; Primes ; Factorization

1- Introduction

Nowadays, there is no general method of factorization of certain numbers. This paper presents an elementary new one.

2. Method fundaments

Let an integer



P

1



K u

1 ₍₁₎

And two primes

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Let L, K, K’ integers and let 1 2 1 2 1 2 ' ' ' '' ' ''' K K u v K K L KK U K K U P P U uvKK U           

The mathematical arguments : We have 2 1 2 (2U L) L 4U K K     2 (2U' L) L 4U' KK'     2 1 2 (2 ''U L) L 4 ''U P P     2 (2U''' L) L 4U''' uvKK'    

Let us make the following computation :

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2 1 2 1 2 1 2 2 2 2 2 1 2 2 1 2 1 2 2 2 2 2 1 2 2 2 2 1 2 1 2 1 2 (2 ' ' 4 ' ). (2( ' '')(2 ' 2 '' 2 ' '') 4( ' ' '' ) (2 ' ' 2 ' ). (2( ' '')(2 ' 2 '' 2 ' '') 4( ' ' '' ) (2 ' ' 2 ' )(2( ' '' ) 2( '' U P P KK U L P P L P P U U U U U U U KK U P P U L P P KK U L P P U U U U U U U KK U P P U L P P KK U L P P U U U P P                       4 2 2 2 1 2 1 2 1 2 3 3 3 2 1 2 1 2 1 2 1 2 2 2 1 2 3 2 2 2 1 2 1 2 1 2 1 2 1 2 3 1 2 1 2 1 ' ')) 4( ' '')( ' ' ' ' 2 ' ' 4 ' 2 ' 4 ' ') 2( '' ' ' '' ' ). (4 ' 2 ' 4 ' ' ' 6 ' 2 ' ')) (( U KK U U U P P U L P P U KK P P U P P KK U L P P U L P P U L P P KK U P P U KK U U U P P U L P P U P P KK LP P KK U L P P L P P U L P P KK A                         2 1 2 1 2 1 2 4 4 3 3 2 2 1 2 1 2 2 1 2 1 2 2 2 2 2 1 2 4 2 2 2 3 1 2 1 2 1 2 1 2 ' ' 4 ' ). ( ' '' 2( '' ' ') ( ' ') ( '' ) ) (2 ' ' 2 ' ). (2( ' '')(2 ' 2 '' 2 ' '') 4( ' ' '' ) 4( ' '') ( ' ' ' ' 2 ' U P P KK U L P P L P P U U U P P U KK U KK U P P U L P P KK U L P P U U U U U U U KK U P P U U L U P P U L P P U KK P P U P                       ₂ 2 2 1 2 4 2 2 2 3 1 2 1 2 1 2 1 2 3 3 2 1 2 1 2 1 2 2 4 4 3 3 2 2 0 1 2 1 2 1 2 1 2 1 2 ')) 2( '' ' ' '' ' ). ( ' ' ' ' 2 ' ' 4 ' 2 ' 4 ' ')) ((2 ' ' 2 ' )( ' '' 2( '' ' ') ( ' ') ( '' ) ) 2( '' P KK U P P U KK U U U P P U L P P U KK P P U P P KK U L P P U L P P U L P P KK A U L P P KK U L P P U U U P P U KK U KK U P P U P P                       2 2 4 2 2 2 3 1 2 1 2 1 2 1 2 ' ' '' ' ) ( ' ' ' ' 2 ' '))) U KK U U L U P P U L P P U KK P P U P P KK       

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2 4 2 2 2 2 2 2 2 4 2 2 2 2 2 1 2 1 ( ' ''' ' ( ' ''') ''') ' (( ' ''') ( ' ''') ( ' ') 2( ' ''') ' ') ( ' ''' ' ( ' ''') ''') ' ''' ' (( ' ''') ( ' ''') ( ' ') 2( ' ''') ' ') '' ( '') 1 U U KK U U U KK uv U U U U L U KK U U U KK U U KK U U U KK U KK U U U U L U KK U U U KK UU P P U U U uv                          1 2 2 1 2 2 2 2 1 2 1 2 1 2 2 1 2 2 1 2 1 2 2 1 2 2 2 2 1 2 1 2 1 2 2 1 2 4 2 2 2 2 ( ) ( '' ( '') ) ( ) ' ( ) ' ' ( ') 1 ( ) ( ' ' ( ') ) ( ) '' ( ) ( ' ''') ( ' ''') ( ' ') 2( ' ''') ' ' ' ( ' P P U P P UU P P U U U P P U P P U P P UU KK P P U U U P P uv U P P UU KK P P U U U P P U P P U P P U U U U L U KK U U U KK U U                           4 2 2 2 2 1 2 1 2 4 2 2 2 2 ') ( '') ( '' ) 2( '') '' '' ( ') ( ') ( ' ') 2( ') ' ' U U L U P P U U U P P U U U U L U KK U U U KK              2 4 2 2 2 2 2 2 2 4 2 2 2 2 2 1 2 1 ( ' ''' ' ( ' ''') ''') ' (( ' ''') ( ' ''') ( ' ') 2( ' ''') ' ') ( ' ''' ' ( ' ''') ''') ' ''' ' (( ' ''') ( ' ''') ( ' ') 2( ' ''') ' ') '' ( '') 1 U U KK U U U KK uv U U U U L U KK U U U KK U U KK U U U KK U KK U U U U L U KK U U U KK UU P P U U U uv                          1 2 2 1 2 2 2 2 1 2 1 2 1 2 2 1 2 2 1 2 1 2 2 1 2 2 2 2 1 2 1 2 1 2 2 1 2 4 2 2 2 2 ( ) ( '' ( '') ) ( ) ' ( ) ' ' ( ') 1 ( ) ( ' ' ( ') ) ( ) '' ( ) ( ' ''') ( ' ''') ( ' ') 2( ' ''') ' ' ' ( ' P P U P P UU P P U U U P P U P P U P P UU KK P P U U U P P uv U P P UU KK P P U U U P P U P P U P P U U U U L U KK U U U KK U U                           4 2 2 2 2 1 2 1 2 4 2 2 2 2 ') ( '') ( '' ) 2( '') '' '' ( ') ( ') ( ' ') 2( ') ' ' U U L U P P U U U P P U U U U L U KK U U U KK              (I)

From the only value of U and knowing L, we have the two primes

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3. Application sample 1 2 1 2

1000

'

''

'

'''

1025 25 1030 30 1040 40 1040.3 40.3

L



K K

 

U

KK



U



PP



U



uvKK



U

















2 2 1 2 1 2 1 2 1 2 2 4 2 2 2 2 1 2 1 2 12 12 1 (( ) ( ''' ) ( 2 ' ' 2( ') )) '( ) (( ') ( ') ( ' ') 2( ') ' ') 0 1 1 (1.682767.10 ) (1.66167.10 ) 0 0.9855769 U P P U U P P UU KK P P U U U P P uv KK U P P P P U U U U L U KK U U U KK uv uv                      2 2 1 2 1 2 1 2 1 2 4 2 2 2 2 4 2 2 2 2 1 2 1 2 8 8

1 ( 2

'

2(

')

2 ''

2(

'')

)

(

')

(

')

( '

')

2(

')

'

(

'')

(

'')

( ''

)

2(

'')

''

0

1

1 (5.668.10 )

5.58625.10

0 0.9855769

UU

KK

P P

U

U P P

UU P P

U

U P P

uv

U

L

U

KK

U

KK

U

L

U

P P

U

P P

uv

 



























 



8 12 12 8

25 (5.668.10 )( (1.66167.10 )) (1.682767.10 )( 5.58625.10 )

0 U









Application

One of the applications of this method concerns Fermat numbers ! Conclusion

Froman elementary calculation of the zeros of a sixth degree polynomial equation, we have introduced a new method of factorization of numbers.

References

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