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- Summary
- We describe the representation of positive integers in the form of the power series ##q2^n##. This allows us to consider positive integers based on the comparison of partial sums of such power series.

Let (

(q2^0 + q2^1+ . . . + q2^x),

(q2^0 + q2^1+ . . . + q2^y),

(q2^0 + q2^1 + . . . + q2^z),

where: q = (1, 2), (x, y, z) = (1, 2, 3, . . ., n).

In the case if and only if

[(q-1)2^0 +(q-1)2^1 + . . . + (q-1)2^x] := \Delta{

[(q-1)2^0 +(q-1)2^1 + . . . + (q-1)2^y] := \Delta{

[(q-1)2^0 +(q-1)2^1 + . . . + [(q-1)2^z] := \Delta{

Indeed, it is obvious that if q = 1, then [(q-1) 2 ^ n] = 0

[(q-1)2^0 +(q-1)2^1 + . . . + [(q-1)2^z] := \Delta{

For example:

23=[2^0+2^1+2^2+2^3+(2^3)], where; \Delta{

40=[2^0+2^1+2^2+2^3+2^4+(2^0+2^3)], where: \Delta{

63=[2^0+2^1+2^2+2^3+2^4+2^5+( 0 )], where: \Delta{

Is this a theorem or a conjecture ?

What does this generally mean ?

Thank you sincerely!

*a, b, c*) be some arbitrary positive integers such that:(q2^0 + q2^1+ . . . + q2^x),

(q2^0 + q2^1+ . . . + q2^y),

(q2^0 + q2^1 + . . . + q2^z),

where: q = (1, 2), (x, y, z) = (1, 2, 3, . . ., n).

In the case if and only if

*q*=*2,*we accept the following notation :[(q-1)2^0 +(q-1)2^1 + . . . + (q-1)2^x] := \Delta{

*a*},[(q-1)2^0 +(q-1)2^1 + . . . + (q-1)2^y] := \Delta{

*b*},[(q-1)2^0 +(q-1)2^1 + . . . + [(q-1)2^z] := \Delta{

*c*},Indeed, it is obvious that if q = 1, then [(q-1) 2 ^ n] = 0

**Polynomial Conjecture.***We assume that: there exist only finitely many positive integers so that*:[(q-1)2^0 +(q-1)2^1 + . . . + [(q-1)2^z] := \Delta{

*c*} = 0,For example:

23=[2^0+2^1+2^2+2^3+(2^3)], where; \Delta{

*a*}:=2^3,40=[2^0+2^1+2^2+2^3+2^4+(2^0+2^3)], where: \Delta{

*b*}:=2^0+2^3,63=[2^0+2^1+2^2+2^3+2^4+2^5+( 0 )], where: \Delta{

*c*}:=0Is this a theorem or a conjecture ?

What does this generally mean ?

Thank you sincerely!