## 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

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$\phi \, (n) \; = \; \phi \, (n+2)$

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

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

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

for    $n \, \leq \, 10^6$

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