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Old 2018-02-27, 12:13   #1
JM Montolio A
 
Feb 2018

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Smile A new aproach to C. Erdös-Straus. 4/n=...

For any n, find some m, and 3 dividers of (n*m), with: A+B+C = 4*m
Proof.

n*m=a*A=b*B=c*C

4/n =(4*m/n*m) =(A+B+C)/(n*m) =(1/a)+(1/b)+(1/c) !

JM M
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Old 2018-02-27, 19:59   #2
JM Montolio A
 
Feb 2018

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Using that, I find:

4/n = 1/a + 1/b + 1/c

n ( a , b , c )
--------------------------------------------------------------
0+2w ( n ,n ,n/2 )
2+3w ( (n+1)/3 ,n(n+1)/3 ,n )
3+4w ( (n+1)/4 ,n(n+1)/4 ,- )
5+8w ( (n+3)/4 ,(n/2)*(n+3)/4 ,n(n+3)/4 )
---------------------------------------------------------------

Happy to share it here.

JM Montolio
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Old 2018-02-27, 20:03   #3
Batalov
 
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Phi(4,2^7658614+1)/2

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Quote:
Originally Posted by JM Montolio A View Post
Proof.
Sweet Mother of the Lord!

Did you decide to solve all of the problems of the century - and all in one day?
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Old 2018-02-27, 20:04   #4
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Quote:
Originally Posted by JM Montolio A View Post
For any n, find some m, and 3 dividers of (n*m), with: A+B+C = 4*m
Proof.

n*m=a*A=b*B=c*C

4/n =(4*m/n*m) =(A+B+C)/(n*m) =(1/a)+(1/b)+(1/c) !

JM M
Divisors for one... And by parity argument we get that an even number of A,B,or C must be odd.
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Old 2018-02-27, 20:12   #5
JM Montolio A
 
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Straüss-Erdös
------------------------------------------------------------
n 2 m 1 nm 2 4m= 1+ 1+ 2 (a 2 b 2 c 1 )
n 3 m 2 nm 6 4m= 1+ 1+ 6 (a 6 b 6 c 1 )
n 4 m 1 nm 4 4m= 1+ 1+ 2 (a 4 b 4 c 2 )
n 5 m 2 nm 10 4m= 1+ 2+ 5 (a 10 b 5 c 2 )
n 7 m 4 nm 28 4m= 1+ 1+ 14 (a 28 b 28 c 2)
--------------------------------------------------------------
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Old 2018-02-27, 20:15   #6
JM Montolio A
 
Feb 2018

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Exclamation

Did "you" decide to solve all of the problems.

"you" not. We. "NOSOTROS".

I'm only sharing knowledge.

JM M
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Old 2018-02-27, 20:20   #7
JM Montolio A
 
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also i find:


N 4m A B C ( a b c)
-----------------------------------------------------
7 4*4= 16 = 7 +7 +2 4/n =( 4 4 2n)
49 4*4= 16 = 7 +7 +2 4/n =( 28 28 2n)
77 4*4= 16 = 7 +7 +2 4/n =( 44 44 2n)
91 4*4= 16 = 7 +7 +2 4/n =( 52 52 2n)
119 4*4= 16 = 7 +7 +2 4/n =( 68 68 2n)
--------------------------------------------------------
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