Find-all-k-m-n-N-such-that-k-3-m-3-n-

Question Number 180695 by depressiveshrek last updated on 15/Nov/22

F i n d a l l k, m, n \in N s u c h t h a t

k! + 3^{m} = 3^{n}

Answered by nikif99 last updated on 16/Nov/22

T r y i n g a n a n s w e r .

m u s t b e m < n a n d 3^{n} m o d 3 = 0 \Rightarrow

(k! + 3^{m}) m o d 3 = 0

i f k ⩾ 7 \Rightarrow [6! \times 7 \times \dots + 3^{m})] m o d 3 = 0 \Rightarrow

(3^{m} m o d 3 = 0) \land [(6! \times 7 \times \dots) m o d 3] = 0

2 n d t e r m i s n o t v a l i d \Rightarrow

k \in {0 . . 6}, s o (k, m, n) =

(2, 0, 1), (3, 1, 2), (4, 1, 3), (6, 2, 6)

Commented by Frix last updated on 16/Nov/22

but k! \mod 3 = 0 for k ⩾ 3 because obviously

with k ⩾ 3 we have k! = 1 \times 2 \times 3 \times 4 \times \dots . = 3 N

do I get something wrong ?

Commented by nikif99 last updated on 16/Nov/22

Y e s, y o u a r e r i g h t . I t n e e d s m o r e

e x a m i n a t i o n .

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