A Special Case (4)

I am going back again to that grid table that lists the first 9 numbers in the unique sequences associated with the equation (x – 1)ⁿ = 0 from n = 1 to 9.

1	1	1	1	1	1	1	1	1
1	2	3	4	5	6	7	8	9
1	3	6	10	15	21	28	36	45
1	4	10	20	35	56	84	120	165
1	5	15	35	70	126	210	330	495
1	6	21	56	126	252	462	792	1287
1	7	28	84	210	462	924	1716	3003
1	8	36	120	330	792	1716	3432	6435
1	9	45	165	495	1287	3003	6435	12870

Now an especially interesting feature with respect to these numbers can be made where n is a prime number. .

If we look at the entry with respect to n = 2, a unique pattern unfolds, whereby numbers follow a cyclical pattern of 2 members, where the 1^st when divided by 2 leaves a remainder of 1, and where 2 is then a factor of the 2^nd number.

Then when we look at the entry with respect to n = 3, another unique pattern unfolds, whereby numbers now follow a cyclical pattern of 3 members, whereby the 1^st when divided by 3 leaves a remainder of 1, with 3 then a factor of the other two numbers.

Finally to again illustrate, with respect to n = 5, another unique pattern unfolds, whereby numbers follow a cyclical pattern of 5 members, whereby the 1st when divided by 5 leaves a remainder of 1, with 5 a factor of the other 4 members of the cyclical group.

Then when n is not prime, no such unique cyclical pattern unfolds.

For example when n = 4, the 1st number (in the first group of 4) leaves a remainder of 1 (when divided by 4). However the 1^st number (in the next group of 4 i.e. 35) leaves a remainder of 3 (when divided by 4).

Also, though 4 for example is a factor of the second member (of the first group of 4) it is not a factor of the 3^rd member.

So again in an “Alice in Wonderland” type manner this provides in principle a valid manner for testing whether any number is prime, which in general terms can be expressed as follows.

With respect to the unique number sequences associated with the polynomial equation,

 (x – 1)ⁿ = 0, for n = 1, 2, 3, 4,…,

if n is prime then a unique cyclical sequence of n individual members will unfold, whereby the 1st member of each cyclical group will always leave a remainder of 1 (when divided by n) with n then being a factor of the other (n – 1) members of the group.

And if this is not the case, then n is not a prime number.

As the unique number sequences involved are potentially infinite, this would then strictly prove a never-ending process.

However if the required pattern for prime numbers applies to the 1st cyclical group of members involved, then it will apply to all further groups.

Therefore we can more simply restate the position as follows.

With respect to the unique number sequences associated with the polynomial equation,

 (x – 1)ⁿ = 0, for n = 1, 2, 3, 4,…,

if n is prime, then a unique cyclical sequence of n individual members will unfold, whereby the 1st member of the first cyclical group will leave a remainder of 1 (when divided by n) with n then a factor of the other (n – 1) members of the group.

For example if 7 is prime, the unique cyclical sequence of 7 individual members will unfold, whereby the 1st member (i.e. 1) of the first cyclical group leaves a remainder of 1 (when divided by 7) with 7 then being a factor of the other 6 members of the group.

And as we can see from the table, these six members are 7, 28, 84, 210, 464 and 924 respectively with 7 a constituent factor of all these numbers.

So 7 therefore is a prime number. (Incidentally the next number, which is then the 1^st member of the next cyclical group i.e. 1716 as expected, then leaves a remainder of 1, when divided by 7).

However 8 clearly is not prime. Whereas the 1^st member - as is always the case - leaves a remainder of 1 (when divided by 8), 8 is not a factor for example of the second i.e. 36 of the remaining seven members of the group.

As I have mentioned on many occasions in these blog entries, there are really two complementary ways of defining a prime number.

The standard (Type 1) approach, based on cardinal number identity, defines a prime as an indivisible “building block” of the natural number system, so that it has no constituent factors (other than itself and 1). This corresponds directly with the quantitative aspect of number.

However the alternative (Type 2) approach based on ordinal number identity defines a prime in reverse as a group that is already composed of natural number members.

So for example 5 as a prime number is thereby necessarily composed of its 1^st, 2^nd, 3^rd, 4^th, and 5^th members.

This directly corresponds with the qualitative aspect of number, which then indirectly can be expressed in quantitative fashion through obtaining the 5 roots of 1.

Now interestingly as one of the n roots of 1 is always 1, this thereby is not unique.

So for a prime number n, the remaining (n – 1) roots are unique.

So what we have uncovered therefore through these unique number sequences of a simple type of polynomial equation, is an approach to the primes that reflects the ordinal (qualitative) rather than cardinal (quantitative) definition of a prime.

So once again from the standard cardinal perspective, a number n is prime if it has no constituent prime factors.

However, from this alternative ordinal perspective, a number n is prime if it is a constituent factor of a unique group of n – 1 members (with the remaining member = 1).  

A Special Case (3)

In the last blog entry, specifically with reference to the number “2”, I showed how there are two equally important ways in which the number can be defined.

Thus in standard linear terms, where its sub-units are considered in an absolute independent manner, 2 is given an analytic quantitative identity (in cardinal terms).

However in paradoxical circular terms, where its sub-units are now considered as relatively interdependent - and thereby fully interchangeable with each other - 2 is given a holistic qualitative identity (in an ordinal manner).

Though the ordinal nature of number is of course recognised in conventional mathematical terms, it is invariably reduced in a merely quantitative manner (whereby each position is given a fixed identity).

Thus using a physical analogy from quantum physics, every number can manifest itself in both a particle and wave-like manner, and these two aspects necessarily keep switching with each other in the dynamics of understanding.

However the conventional mathematical approach grossly misrepresents the true nature of number by attempting to (formally) interpret it in a static fixed manner.

Now, I have frequently referred to the two aspect of number as Type 1 and Type 2 respectively!

Type 1 corresponds with the linear conventional interpretation of number in analytic quantitative terms.

The natural numbers from this perspective (1, 2, 3, 4, 5, …) are more fully represented as   

1¹, 2¹, 3¹, 4¹, 5¹, …

Type 2 corresponds (indirectly) with the circular paradoxical interpretation of numbers in a holistic qualitative manner

The natural numbers from this perspective are more fully represented as

1¹, 1², 1³, 1⁴, 1⁵, …

Now these have no distinctive meaning in quantitative terms. However from a holistic qualitative perspective 1² represents the interdependence of two related units.

Then indirectly this interdependence can be expressed in a holistic circular manner as + 1 and  – 1, where both units are interchangeable.

(Of course + 1 and – 1 can equally be expressed - as in conventional mathematical terms - in an absolute manner where they are clearly separated in an analytic manner).

However it is the continued failure to recognise the true holistic aspect of number, which I am addressing in these blogs.

So mathematical understanding necessarily contains both analytic and holistic aspects, which in direct terms are represented by reason and intuition respectively.

For example I have just read this quote from Poincare.

“It is by logic, we prove, by intuition we invent”

Logic - in the sense that Poincare intends - represents the analytic aspect of understanding which is of an unambiguous rational nature.

However intuition represents the corresponding holistic aspect, which inherently is of a paradoxical circular nature.

Therefore though analytic reason and holistic intuition are clearly distinct from each other, in conventional mathematical terms the holistic aspect is inevitably reduced in formal terms to the analytic.

In this sense formal mathematical interpretation is of a grossly reduced nature (and ultimately not fit for purpose).

But before this crucial point can be properly grasped, the distinctive holistic aspect of all mathematical understanding must be properly recognised.

Now if we go back to the original simple expression i.e. x = 1, we can perhaps now better appreciate what happens when both sides are raised to the power of n.

So xⁿ = 1ⁿ.

Therefore when n = 2, x² = 1².

Therefore, the number on the right side of the equation properly belong to the Type 2 aspect of the number system.

However in conventional mathematical terms, as 1² has no distinctive quantitative value, it is inevitably reduced to 1 (i.e. 1¹) where it is interpreted as the first natural number in the Type 1 system.

Thus a number that should be treated in a holistic manner - and indirectly expressed in a circular interdependent fashion - is now treated in analytic terms.

And this is what then creates the conflict with earlier analytic type interpretation.

So once again when x = 1, x – 1 = 0.

Therefore squaring both sides (x – 1)² = 0.

And this equation in Type 1 terms has two “linear” solutions i.e. + 1 and + 1 (as independent).

However equally when x = 1, x² = 1² .

And this equation in Type 2 terms has two “circular” solutions i.e. + 1 and – 1 (as interdependent).

Thus we obtain two “different” answers because both correspond respectively to different notions of dimensions (that are analytic and holistic with respect to each other).

However again, these two sets of answers cannot be reconciled satisfactorily through conventional mathematical interpretation (that solely recognises the analytic aspect of number as quantitative).  

A Special Case (2)

In the last entry, I started with the sample identity i.e. x = 1, to generate the corresponding polynomial equation (degree 1) i.e. x – 1 = 0.

Now in conventional mathematical terns both of these, i.e. x = 1 and

x – 1 = 0 would be treated as identical.

However something strange occurs when we square both expressions.

Therefore in the former case, we obtain x² = 1 so that x² – 1 = 0, whereas in the latter we obtain (x – 1)² = 0, so that x² – 2x + 1 = 0.

So two different equations are now derived from squaring the two - apparently - similar expressions.

As we know the two roots of (x – 1)²  = x² – 2x + 1 = 0, are + 1 and + 1 respectively and the last blog entry was designed to explain how, from using the unique number sequence associated with this equation, we can in fact properly distinguish these two linear - and indeed in more general terms - n repeated examples of 1.

However, the corresponding roots of x² – 1 = 0 are + 1 and – 1 (which lie as two equidistant points on the unit circle (in the complex plane).

And in more general terms the n roots of xⁿ – 1 = 0, will lie as n equidistant points on the same unit circle.

So we need to explain how we have thereby moved from a linear based notion of dimension (as points on the real number line), to a corresponding circular based notion of dimension (as points on the unit circle).   

Now just as there is a unique number sequence associated with (x – 1)ⁿ = 0, equally there is a unique number sequence associated with xⁿ – 1 = 0 , for every integer value of n.

So in the case of x² – 1, where the coefficient of the x term is 0, again we start with 0,

1. Then we obtain (0 * 1) + (1 * 0) to get 0 as the next term.

So now we have 0, 1, 0.

Then for the next term we obtain (0 * 0) + (1 * 1) = 1.

So the sequence is now 0, 1, 0, 1.

And it continues in this fashion continually alternating between 0 and 1.

In the case of x³ – 1, the coefficients of both the x² and x terms is 0 and we start in this case with 0, 0, 1.

Then to generate the next term we obtain (0 * 1) + (0 * 0) + (1 * 0) = 0.

So we now have 0, 0, 1, 0.

And for the next term we obtain (0 * 0) + (0 * 1) + (1 * 0) = 0. The sequence is now

0, 0, 1, 0, 0.

Then the next term = (0 * 0) + (0 * 0) + (1 * 1) = 1.

Thus the sequence is now 0, 0, 1, 0, 0, 1.

It then continues on indefinitely in this manner with a 1 always followed by two 0’s.

And for the general case, where xⁿ – 1 = 0, in the unique number sequence associated, each 1 is followed by (n – 1) 0’s.

So, as in the previous case for (x – 1)ⁿ = 0, I gave the first 9 digits of the unique digit sequence, for n = 1 to 9, likewise I will now do the same with respect to xⁿ – 1 = 0 (again from n = 1 to 9).

1	1	1	1	1	1	1	1	1
1	0	1	0	1	0	0	0	0
1	0	0	1	0	0	1	0	0
1	0	0	0	1	0	0	0	1
1	0	0	0	0	1	0	0	0
1	0	0	0	0	0	1	0	0
1	0	0	0	0	0	0	1	0
1	0	0	0	0	0	0	0	1
1	0	0	0	0	0	0	0	0

  

In the previous case for (x – 1)ⁿ = 0, we were able to successfully use the unique number sequence associated with the polynomial equation in question (for any required value of n) to approximate the required roots (as solutions to the equation).

So again in the case of (x – 1)² = 0, where the unique digit sequence is 1, 2, 3, 4, 5, …the ratio of n/(n – 1) approximates to 1 thereby giving the first root of the equation.

However in the case where n = 2, x² – 1 = 0 is associated with the unique digit sequence 1, 0, 1, 0, 1, …. . Thus when we attempt to calculate a root we face an - apparently - meaningless situation. For the ratio of the n_th/(n – 1)_th term gives either 1/0 = ∞ or 0/1 = 0.

Thus we generate two conflicting answers ∞ and 0 which are at opposite extremes of each other. Likewise in neither case do we obtain the true roots + 1 and – 1, which in conventional mathematical terms are understood to satisfy the equation.

Now the reason for this problem is that we have in fact now moved from a linear interpretation, which applies in the case of (x – 1)² = 0, to a circular interpretation of number, which applies in the case of  x² – 1 = 0.

The implications of this could not be more important for Mathematics as it implies that two distinctive logical systems are in fact required.

The linear (Type 1) system is based on the standard either/or logic whereby opposite signs are clearly separated in an absolute manner.

However the circular (Type 2) system is based on a paradoxical both/and logic whereby opposite signs are viewed as complementary with each other in a merely relative manner.

From the standard linear perspective, a cardinal number is defined as composed of independent homogeneous units (that lack qualitative distinction).

So 2 = 1 + 1.

However from the paradoxical circular perspective, 2 is now seen as composed of two units that are fully interdependent with each other in a qualitative manner.

This in fact corresponds with the ordinal definition of number of 1^st and 2^nd units.

Thus 2 = 1^st + 2^nd

However in conventional mathematical interpretation standard, these units are given a fixed identity, whereby they can be reduced in cardinal terms.

So again 2 = 1 + 1 (in cardinal terms). 2 = 1^st + 2^nd (in ordinal terms).

So by identifying 1^st as the last unit of 1, and 2^nd then (with the 1st unit fixed) as the last unit of 2, 1^st and 2^nd are thereby reduced in an analytic independent manner to cardinal interpretation.

However when we view the ordinal units as truly interdependent in a relative manner, this requires a new holistic interpretation of 2, whereby 1^st and 2^nd units are understood as fully interactive with each other.

And this requires moving from a linear either/or logic (where opposite signs are viewed as absolutely distinct) to a circular both/and logic (where opposite signs are viewed as fully complementary in a relative manner).

As I have illustrated repeatedly in previous blog entries, we can illustrate the two types of logic very well in the case of the definition of left and right turns at a crossroads.

From the linear logical perspective, a left or right turn would be absolutely designated as + 1 (a left turn) and – 1 (a right turn i.e. not a left  turn).

So when one approaches the crossroads from just one direction left and right turns can be unambiguously identified in a linear logical manner.

So for example when approaching the crossroads while heading N, left and right turns can be unambiguously identified.

Equally when approaching the crossroads while heading S, left and right turns can be unambiguously identified.

So the understanding in both cases concurs with linear logic (where opposite turns are absolutely separated). Quite literally this represents 1-dimensional logic, where just one point of reference is required (e.g. heading N) to unambiguously make an identification.

However if one now envisages the approach to the crossroads from both N and S directions simultaneously - in what now represents 2-dimensional logic - what is a left turn from one direction is right from the other and vice versa. So we have now moved to a new holistic circular interpretation entailing the complementarity of opposite signs. So what is + 1 from one direction (N), is equally – 1 from the opposite direction (S) and what is + 1 from this other direction (S), is equally – 1 from the opposite direction.

Thus positive and negative signs have an absolute fixed interpretation in terms of standard linear (1-dimensional) interpretation.

However positive and negative have a relative interchangeable interpretation in terms of  holistic interpretation (with the example here relating to the 2-dimensional case).

In more general terms, holistic interpretation in this context applies where n (as the number of dimensions involved) > 1.

However what is truly remarkable is that there is no formal mathematical recognition whatsoever for this latter type of circular understanding (where the interdependence of the various sub-units of each cardinal number is recognised).

Rather, in every context, the notion of holistic interdependence is formally reduced in an analytic independent manner.

In other words the Type 2 qualitative notion of number is invariably reduced in a Type 1 quantitative  manner.

However, the unique number sequences associated with xⁿ – 1 = 0, (where n > 1) require true holistic understanding to properly interpret the ratios (as corresponding roots of the equations).

Thus in the case of x² – 1 = 0, the respective ratios of 1/0 and 0/1 i.e. ∞ and 0 lie at opposite extremes of each other as complementary opposites reflecting the holistic interdependence of the roots involved.

Therefore the fact that both + 1 and – 1 are the respective roots in this case indicates that their relationship should be understood in a relative holistic manner (as interchangeable) in the context of the unique number sequence involved.

However once again, though these circular roots are of course recognised in conventional terms as valid solutions to the equation, they are interpreted merely in an analytic manner (where positive and negative signs remain fixed).  

A Special Case (1)

If  x = 1, then we can thereby obtain the simple polynomial equation x – 1 = 0.

The corresponding unique sequence of digits associated with this equation can then be obtained as follows.

Starting with 1, we multiply the negative of the coefficient of the x^{n– 1} term (in this case x⁰ = 1) by 1. Therefore we multiply 1 * 1 = 1 to obtain the 2^nd term.

And as this equation (of degree 1) involves a 1-step procedure, we keep in turn multiplying 1 by 1 to obtain each additional term.

So the unique infinite digit sequence associated with the equation (i.e. x – 1 = 0) is

1, 1, 1, 1, 1,…

Now if we obtain the square of our expression then (x – 1)² = 0, i.e.

x² – 2x + 1 = 0.

Now to calculate the unique digit sequence associated with this equation, we start with

0, 1 and then obtain (2 * 1) – (1 * 0) = 2 + 0 = 2. 

So we now have 0, 1, 2.

Then the next term = (2 * 2) – (1 * 0) = 3.

And by continuing in this manner we find that the unique digit sequence associated with the equation (x² – 2x + 1 = 0) is the set of natural numbers i.e.

1, 2, 3, 4, 5, 6,….

However there is an alternative way of obtaining this sequence from the previous sequence, whereby the nth term in the latter represents the sum of the first n terms in the previous sequence.

And this procedure can be extended indefinitely.

Thus when we cube (x – 1)³ = 0, i.e. x³ – 3x² + 3x – 1 = 0, the unique digit sequence associated with this equation can be directly obtained from the previous sequence, whereby once again the nth term of the latest represents the sum of the first n terms of the previous sequence.

So the unique digit sequence associated with the equation x³ – 3x² + 3x – 1 = 0 is

1, 3, 6, 10, 15, 21,…

And to give just one more illustration the unique digit sequence for (x – 1)⁴  = 0 

i.e. x⁴ – 4x³ + 6x² – 4x + 1 = 0 is

1, 4, 10, 20, 35, 56,…

In the following grid, I show the first 9 terms in the respective unique number sequences for the 9 equations from x – 1 = 0 to (x – 1)⁹ = 0 .

1	1	1	1	1	1	1	1	1
1	2	3	4	5	6	7	8	9
1	3	6	10	15	21	28	36	45
1	4	10	20	35	56	84	120	165
1	5	15	35	70	126	210	330	495
1	6	21	56	126	252	462	792	1287
1	7	28	84	210	462	924	1716	3003
1	8	36	120	330	792	1716	3432	6435
1	9	45	165	495	1287	3003	6435	12870

Now the remarkable feature regarding these number sequences is that there is a direct horizontal/vertical type correspondence as between entries.

In other words, the sequence of numbers in each row (read from left to right) exactly matches the sequence of numbers (read from top to bottom) in the corresponding column.

And when interpreted properly there is a fascinating explanation for this significant finding.

As we have seen, 1 represents the solution to all these equations.

However whereas in the first case (i.e. for x – 1 = 0), this solution occurs just once, in all other cases it does so on multiple occasions, so that for example for (x – 1)⁹ = 0, it repeats 9 times.

Now whereas in conventional mathematical terms, there is no distinction as between these separate solutions, in fact this is not quite the case (with each root or solution for x carrying a unique meaning).

This can best be illustrated with respect to the 2-dimensional situation, whereby 1 - as the solution for x - occurs twice.

Now if we think of this in geometrical terms as a 2-dimensional figure e.g. a square, it has two sides i.e. length and width.

However when length and width = 1, there is a distinction in that these two measurements are horizontal and vertical with respect to each other.

Thus if we identify the length with the horizontal measurement, then the width represents the vertical (and vice versa).

This is equally true in 3 dimensions. So if we now add in the height (in a cube with each side = 1), it is now vertical with respect to each of the other measurements (considered as horizontal). And though this cannot be visualised with respect to higher linear dimensions, the same principle holds with each new dimension distinguished as vertical in relation to any other dimension considered as horizontal.

And we can see that this property is intrinsic to the very nature of the number sequences associated, with the equations in the various dimensions having a distinctive horizontal/vertical correspondence in evidence.

So this enables one to distinguish each new unitary solution to x (as the degree of the equation increases by 1) in a distinctive manner (as relatively vertical to any previous root representing a horizontal measurement).    

Polynomial Equations and Associated Number Sequences (2)

In the last entry I mentioned that the unique number sequence associated with a given polynomial equation can be used to approximate the real roots of that equation.

When there is more than one real solution to the equation, the initial root approximated from the unique number sequence will relate to the largest valued root (in absolute terms).

For example, to illustrate, let us take the simple binomial equation x² + 2x – 3 = 0.

Then starting with 0, 1 we add (– 2 * 1) + (3 * 0) = – 2, which is then the next term in the unique number sequence of this equation.

So we now have 0, 1, – 2.

We now add  (– 2 * – 2)  + (3 * 1) = 7.

This gives us 0, 1, – 2, 7.

Continuing again in this manner, we obtain the next number in the sequence as

(– 2 * 7) + (3 * – 2) =  – 20.

So the unique number sequence is now 0, 1, – 2, 7, – 20.

Even with these few terms we can approximate the largest root (ignoring the sign) as

–        20/7 = – 2.857…

Thus it already appears that the value of this root is quickly converging to – 3.

Now of course, because this is a binomial equation, this implies that the other root is also real and in this case as the product of the two roots = – 3 (as the value of c in the general form of the equation) the other root therefore =  1.

However there is a procedure for approximating the other real roots, which could be useful for polynomial equations of degree > 2).

So we multiply the 1^st hidden term of the sequence i.e. 0 by – 3 = 0. We then subtract this from the next term 1 to obtain 1, which is then the 1st term of a new number sequence..

We then multiply the next term of the original sequence (i.e. 1) =  – 3 and then subtract this from the next term in the original sequence (i.e. – 2).

Thus we obtain – 2 – (– 3) = – 2 + 3 = 1.

So 1 is again the 2^nd term in the new number sequence.

Continuing on, we multiply the 3^rd term of the original series (i.e. 7) again by the estimated root (i.e. – 3) = – 21. So we then subtract this from the next number in the original series (i.e. – 20) to obtain – 20  – (– 21) = – 20 + 21 = 1.

So 1 is again the 3^rd term in the new sequence.

And if we continued on in this manner, 1 will always be the next number generated in the new number sequence.

So now with respect to this new number sequence 1, 1, 1,… the estimated value of the second root = 1/1/ = 1.

Thus the two roots are – 3 and 1.

And though we have illustrated this with respect to two real roots in principle the same approach can be extended for any number of real roots.

Of course the values of the roots will not always exist as simple rational numbers.

However, where irrational values are involved, their value can in principle be approximated to any required degree of accuracy by calculating a sufficient amount of terms in the unique number sequence (associated with the polynomial equation in question) and then applying the procedures illustrated.

We can also reverse the approach so that starting with a given sequence of numbers we can then attempt to construct the unique polynomial equation to which they relate.

However for a meaningful finite equation to result it is important that this sequence of digits to be ordered in a consistent manner. Where this is the case, a finite number of steps is required to generate the digits in question.

However when not properly ordered one can keep generating potentially an unlimited number of steps in the attempt to match the given sequence of digits.

So I will briefly illustrate here for both cases.

The first sequence relates to what are known as the Pell numbers, i.e.

1, 2, 5, 12, 29, 70, 169,…

Now initially - even if it is in fact an ordered sequence - we do not know how many terms are involved.

However seeing as the next term (after 1) is 2 then this means that the coefficient of the

x^{n – 1} term is – 2. So we now multiply 2 by 2 and see what adjustment has to made to derive 5 (which is the next term in the sequence). So clearly we need to add 1 in this case, which implies that the coefficient of the x^{n – 1} term is – 1. So we now combine (5 * 2) + (2 * 1) = 12 indicating that no adjustment using coefficients of further terms is necessary.

So n = 2 in this case, so that the equation which uniquely generates the Pell numbers is x² – 2x – 1 = 0.

It bears a relationship to the Fibonacci equation, except that in this case we keep combining twice the last term in the sequence derived with the previous term!

As the positive real-valued root of this equation = √2 + 1, we can use its associated unique number sequence to approximate √2.

So taking the last two terms the ratio is 169/70 = 2.414285…

Therefore the approximation for √2 = 1.414285… (which is already correct to the first 4 significant figures).  

However the well known sequence of odd integers, i.e. 1, 3, 5, 7, 9,…is not so well ordered in this case.

The coefficient of the x^{n – 1} term is – 3. Then we multiply 3 by 3 and add it to x * 1 to obtain the next term in the sequence i.e. 5. Therefore x = – 4, so that the coefficient of the  x^{n – 2} term is 4.

Then (3 * 5) + (– 4 * 3) + (x * 1) = 7 (as the next term).

This entails that x = 4, so that the coefficient of the x^{n – 3}  term is – 4.

If we stopped here, we would get the equation

x³ – 3x² + 4x – 4.

Now, we know that the ratio of the n_th to the previous term for the odd integers converges to 1. However strictly when we divide these numbers there will always be a remainder of 2.

It is fascinating therefore in this context that when we substitute 1 in the equation (of degree 3) above that we get an answer of – 2.

And in absolute terms we will always obtain the same number no matter how many terms we include in the equation.

In fact a definite pattern emerges whereby the coefficient of each successive term keeps alternating as between + 4 and – 4. Therefore when we substitute 1 in the polynomial equation involved the answer in absolute terms = 2 (continually alternating as between + 2 and – 2).

This is a pattern (i.e. alternating as between + and –) that - as we will see - is directly associated with imaginary numbers.

In fact if we take the earlier binomial expression of the odd number integer sequence, it will be given as x² – 3x + 4 = 0.

When we let x = 1 (as indicated by the ratio of successive higher numbered terms), we get the result of + 2.

And there are no real valued solutions to this equation, with the two solutions imaginary.  

However in the case of the odd integers there is clearly a discernible pattern to the numbers.

However there are a great many possible infinite number sequences which offer no discernible pattern and therefore cannot be successfully associated with polynomial equations of a finite size.   

Polynomial Equations and Associated Number Sequences (1)

As is well known the Fibonacci sequence is

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987,…

This in turn is related to the simple binomial equation, x² – x – 1 = 0.

The positive real valued solution to this equation is given by the important constant

ϕ (phi) = (1 + √5)/2  = 1.618033….

The Fibonacci sequence itself can be obtained in the following manner.

The general binomial equation can be given as ax² + bx + c = 0.

Where a = 1, this simplifies to x²+ bx + c = 0.

Then starting with the two numbers 0 and 1 we can obtain the Fibonacci sequence in the following manner.

To obtain the next number add 1 *(– b) + 0 (– c).

So in this specific case we get 1 * (1) + 0 * (1)  =  1.

Thus the next term in the Fibonacci sequence = 1.

So we now have the terms 0, 1, 1.

Then to obtain the next term we again multiply this latest generated term (i.e. 1) by – b, before adding it to the previous terms (i.e. 1) multiplied by – c.

So we thereby obtain in this case 1 * (1) + 1 * (1)  =  2.

We now have the terms 0, 1, 1, 2.

Thus once more to generate the next term we again multiply the latest term generated (i.e. 2) by – b, before again combining it with the previous term (i.e. 1) multiplied by – c.

This now gives 1 * (2) + 1 * (1)  =  3 which is the next term in the sequence.

Therefore, continuing on in this fashion, we can generate as many terms as we wish with respect to the infinite Fibonacci sequence.

Now the interesting thing is that we can then approximate the value of ϕ (phi) by dividing the latest term in the sequence by the previous term or in more general terms the

n_th/(n – 1)_th term.

Now if we attempt this from the limited number of terms already generated i.e.

0, 1, 1, 2, 3 we will get 3/2 = 1.5 which is not a very good approximation to the true value 1.618033…

However when we divide the 16^th term listed above (i.e. 987) by the previous term (610) we obtain 1.618032… which is already correct to 5 decimal places!

So the approximation improves rapidly through using later terms in the series.

However, though the use of this approximation approach is indeed well known with respect to the Fibonacci equation (and corresponding sequence), what is not equally emphasised is that this same general procedure can in principle be used with respect to any polynomial equation (with one variable).

Thus associated with every polynomial equation (with one variable) is a unique infinite sequence of digits, which can then be used to approximate all the real valued solutions - if indeed such solutions exist - for the equation in question.

Now in applying this approach xⁿ, the term of highest degree should be expressed with unitary coefficient.

Therefore, for example the equation 3x² + 2x – 5 = 0, would be expressed as

x² + 2/3x – 5/3 = 0, before continuing in the previous manner.

Indeed one interesting class of equation relates to the extension of the Fibonacci equation to dimensions (> 2).

For example the Tribonacci equation - where the highest term is of degree 3 - would be given as

x³ – x² – x – 1 = 0.

Now the general equation for equations of degree 3 would be expressed as

x³ + bx² + cx + d = 0.

Then to generate the corresponding Tribonacci sequence we now start with the 3 numbers 0, 0, 1 (Though the two 0’s do not strictly comprise terms in the sequence, they are necessary so as to generate initial terms in the sequence in a correct manner.

And the correct number of 0’s involved will always be 1 less that the highest power of x in the polynomial equation!)

So in this case we add {– b * (1)} + {– c * (0)} + {– d * (0)}  which in the case of the Tribonacci equation = {1 * (1)} + {1 * (0)} + {1 * (0)} = 1.

Thus the sequence is now 0, 0, 1, 1,

Continuing on in this manner, the next term = {1 * (1)} + {1 * (1)} + {1 * (0)} = 2.

We now have 0, 0, 1, 1, 2,

Then the following term = {1 * (2)} + {1 * (1)} + {1 * (1)} = 4.

So we now have 0, 0, 1, 1, 2, 4,

And if continued to generate fresh terms, we would obtain the Tribonacci sequence

0, 0, 1, 1, 2, 4, 7, 13, 24, 44, 81, 149, 274, 504, 927,…

And once again the real valued solution to the Tribonacci equation i.e. 1.839286… can be approximated through calculation of the n_th/(n – 1)_th term.

So taking the last two terms of the sequence above, this would give 927/504 = 1.839285… which again gives the true result correct to 5 decimal places.

And of course this Fibonacci type procedure can be extended to any required value of n.

So on general we have xⁿ – x^{n – 1} – …– x – 1 = 0.

Where n = 4, we get the Tetranacci equation (with associated Tetranacci numbers in the associated sequence).

Where n = 5 we get the Pentanacci, with 6 the Hexanacci, with 7 the Heptanacci, with 8 the Octonacci and with 9, the Nonacci equations and associated numbers respectively.

And in generating all these number sequences we would keep adding 4, 5, 6, 7, 8 and 9 numbers respectively to generate the next number in the corresponding sequence.

One fascinating use of the positive real valued solutions to these equations (which can be conveniently approximated through the associated number sequences) is in providing approximations to the sum of the Riemann zeta function (for positive integer values).

See Fascinating Connections.