Depth expansion of angle conjecture

Give any positive integer n, if n is divisible by a, it becomes N/A, if it is divisible, multiply it by b plus c (that is, bn+c). Repeat this operation, after a limited number of steps, will you definitely get D?

There are only three answers to this question: 1. It's not necessarily 2. Definitely not three. I'm sure they are all.

The following is all about a certain capital.

A = B = C = D = M。

Two a=m b= 1 c=- 1 d=0.

Three a=m b=c=d= 1

Four A = two B = two M- 1 C =- 1 D = 1

Above (m> 1)

Five a = 2 b = 2m-1c =1d =1

6 A = 2 B = C = D = 2M- 1

M above is an arbitrary natural number.

The simplest example:

a=b=c=d=2

a=2 b= 1 c= 1 d= 1

a=2 b= 1 c=- 1 d=0

There are only five original questions. When m=2, it is said that many people in China will prove that the original problem is only a very small part.

All the above data are true, and there is not a counterexample. This question is very short, but it implies very rich mathematical ideas ... there are many things to be used. Those theorems and formulas are perfect and can express very common mathematical laws. This is a mathematical problem, not a guess. This topic focuses on cultivating students' independent thinking ability and reverse thinking. ...

Actually, this question is very simple.

I don't know if this is a holistic approach.

The first step in the overall proof of the above situation:

Firstly, a 2 yuan function is constructed, which reveals a secret: all natural numbers divisible by A are converted into natural numbers f(x, y) that are not divisible by A..

Five a = 2 b = 2m-1c =1d =1

Decomposition of natural numbers by mathematical induction and division ... Prove:

(2^(mn)- 1)/(2^n- 1)=e

When m and n are natural numbers, e is odd.

m= 1 A 1=( 1)

m=2 A2=( 1，5)

m=3 A3=( 1，9， 1 1)

m=4 A4=( 1， 17， 19，23)

m=5 A5=( 1，33，35，37，39)

m=6 A6=( 1，65，67，7 1，73，79)

...

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All terms in the general term formula of the combined infinite sequence A () cannot be divisible by the m power of 2-1.

This combination sequence is very simple, just the first item of countless arithmetic progression. ....