Equations φ(n+1)=φ(n); φ(n+2)=φ(n), and φ(n+3)=φ(n) | φ(n)=φ(n+1)=φ(n+2)

 
 
\phi \, (n) \; = \; \phi \, (n+1)        for   n \; \leq \; 10^4

the solutions n = 1, 3, 15, 104, 164, 194, 255, 495, 584, 975, 2204, 2625, 2834, 3255, 3705, 5186, 5187

\phi \, (1) \; = \; \phi \, (2) \; = \; 1
\phi \, (3) \; = \; \phi \, (4) \; = \; 2
\phi \, (15) \; = \; \phi \, (16) \; = \; 8
\phi \, (104) \; = \; \phi \, (105) \; = \; 48
\phi \, (164) \; = \; \phi \, (165) \; = \; 80
\phi \, (194) \; = \; \phi \, (195) \; = \; 96
\phi \, (255) \; = \; \phi \, (256) \; = \; 128
\phi \, (495) \; = \; \phi \, (496) \; = \; 240
\phi \, (584) \; = \; \phi \, (585) \; = \; 288
\phi \, (975) \; = \; \phi \, (976) \; = \; 480
\phi \, (2204) \; = \; \phi \, (2205) \; = \; 1008
\phi \, (2625) \; = \; \phi \, (2626) \; = \; 1200
\phi \, (2834) \; = \; \phi \, (2835) \; = \; 1296
\phi \, (3255) \; = \; \phi \, (3256) \; = \; 1440
\phi \, (3705) \; = \; \phi \, (3706) \; = \; 1728
\phi \, (5186) \; = \; \phi \, (5187) \; = \; 2592
\phi \, (5187) \; = \; \phi \, (5188) \; = \; 2592

 

It follows that the least number   n   which satisfies the equation

\phi \, (n) \; = \; \phi \, (n+1) \; = \; \phi \, (n+2)   is   5186

\phi \, (5186) \; = \; \phi \, (5187) \; = \; \phi \, (5188) \; = \; 2592
 

It is not known whether there exist infinitely many integers   n   for which
\phi \, (n) \; = \; \phi \, (n+1)

For   n \; \leq \; 2 \; \times \; 10^8    there are 391 solutions

 
 
                         **************************************                

 
\phi \, (n) \; = \; \phi \, (n+2)

For   n \; \leq \; 4 \; \times \; 10^6,   there are 7998 solutions

For   n \; \leq \; 100,   the solutions   n   are :   4, 7, 8, 10, 26, 32, 70, 74

\phi \, (4) \; = \; \phi \, (6) \; = \; 2
\phi \, (7) \; = \; \phi \, (9) \; = \; 6
\phi \, (8) \; = \; \phi \, (10) \; = \; 4
\phi \, (10) \; = \; \phi \, (12) \; = \; 4
\phi \, (26) \; = \; \phi \, (28) \; = \; 12
\phi \, (32) \; = \; \phi \, (34) \; = \; 16
\phi \, (70) \; = \; \phi \, (72) \; = \; 24
\phi \, (74) \; = \; \phi \, (76) \; = \; 36

 
                         **************************************                

 

 
The equation  \phi \, (n) \; = \; \phi \, (n+3)   has two solutions,   n = 3   and   n = 5

for    n  \, \leq  \, 10^6

\phi \, (3) \; = \; \phi \, (6) \; = \; 2
\phi \, (5) \; = \; \phi \, (8) \; = \; 4

 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

math grad - Interest: Number theory
This entry was posted in Number Puzzles and tagged . Bookmark the permalink.

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