Rules relating to addition of exponential powers in n-dimensional space

- Ramanraj K

1^st May, 2020

Introduction

A few rules relating to addition of exponential powers in n-dimensional space are given here. The power or exponent of an integer corresponds to the dimensional space it occupies. The traditional distinction maintained between arithmetic and geometry, viewing both as different branches of mathematics, has needlessly resulted in considerable confusion. It is proposed to illustrate that there is no distinction between arithmetic and geometry, and a unified view of both are required to understand dimensions in mathematics that is the language used to represent values in real space. The chief cause for the division has been that, though dimensional space is naturally associated with integers, it has been ignored for sake of convenience in arithmetic. The convenience has been at a great cost. The rules relating to addition of exponential powers make it necessary to view integers strictly along with its exponent and this also makes clear that there is no distinction between arithmetic and geometry as such. The identity bⁿ + (b-1)bⁿ = b⁽ⁿ⁺¹⁾ is critical to understanding higher dimensions, and the same is elaborated here.

The basic rules governing addition of exponential powers along with elementary proofs to Pythagoras Theorem, Fermat's Last Theorem and Beal's Theorem are also discussed. The Pythagorean Triples connect positive integers and right angle triangles and have practical application in construction and many areas of mathematics. It would be useful to state the connection as a theorem. One of the objects of this listing of basic rules is to enable visualization of integers along with their exponents in real space.

Rule 1: All integers have an exponent, and the exponent of a given integer corresponds to the dimensional space it occupies in cartesian coordinate space.

For example, the integer 1 with exponents 1, 2 and 3 correspond to the first three dimensions as follows:

Integer with exponent/dimension	Arithmetic value	Geometric space
11	1	Length
12	1 x 1	Area
13	1 x 1 x 1	Volume

The integer 1 with exponent 1 would refer to length in one dimensional space. The integer with exponent two, would refer to square area in two dimensional space. The integer with exponent three refers to cubic space occupied by the said integer in three dimensional space.

Rule 2: An integer in a higher dimension may be represented in lower dimensions

Integers may be expressed in lower dimensions :

Integer with exponent/dimension	Arithmetic value	Geometric space
11	11	Length
12	11 x 11	Area
13	11 x 11 x 11	Volume

If a number is required to be converted or expressed in lower dimensions, it may be done using chain-link conversion, as numbers may refer to physical quantities, and need to be expressed in lower dimensions without reducing the original value assigned. For instance, 1³ may be converted only as follows:

Integer with exponent/dimension	Arithmetic value	Geometric space
13	13	Volume
13	11 x 12	Volume = Height x Area
13	11 x 11 x 11	Volume = Length x Length x Length

Therefore, 1¹ ≠ 1² ≠ 1³ ≠ 1ⁿ. It is currently maintained that the "powers of one are all one: 1ⁿ = 1."¹ Each number base and its exponent must be dealt with together and cannot be ignored. For example, 1 m of fine gold thread is not equal to 1 sq. m. of gold foil, and both are not equal to a 1 m cube of solid gold.

Rule 3: Pythagoras Theorem:

If a and b are lengths of the sides of a right angle triangle, and c is the length of its hypotenuse, then a² + b² = c²

Proof:

The traditionally used elementary proof takes a right angle triangle ABC, such that AC is the hypotenuse with AB and BC as its sides, and a perpendicular is drawn from the vertex of the right angle to the hypotenuse AC and the point it meets AC is marked as D. Then,

As triangles ADB and ABC are similar,

AD/AB = AB/AC

AD*AC = AB²

Also, as triangles BDC and ABC are similar,

CD/BC = BC/AC

CD*AC = BC²

Adding both equations,

AD*AC + CD*AC = AB² + BC²

AC(AD + CD) = AB² + BC²

AC*AC = AB² + BC²

Therefore, AC² = AB² + BC²

Rule 4: Pythagorean Triples Theorem: If a, b and c are integers and a² + b² = c², then a, b and c correspond to the sides of a right angle triangle such that c is the hypotenuse with a and b as sides.

There is one to one correspondence between integers a, b and c and the points in the Euclidean plane of Cartesian coordinates, as the x-axis and y-axis are perpendicular to each other forming a right angle triangle at the vertex of coordinates.

Rule 5: Repeated Replication Theorem:

Exponentiation is repeated multiplication of the base, and corresponds to symmetrical replication of the base along the axes in n-dimensional space. If b is the base, then its power or exponent n corresponds to dimensional space n extended by symmetrical replication of the base b times along the axes. This addition of exponents in nDspace could be expressed by the following exponential identity, that may be called the Repeated Replication Theorem of exponential powers:

bⁿ + (b-1)bⁿ = b⁽ⁿ⁺¹⁾

Proof:

bⁿ + (b-1)bⁿ

= bⁿ + b⁽ⁿ⁺¹⁾ - bⁿ

= b⁽ⁿ⁺¹⁾

Lemma: In the term (b-1)bⁿ, if (b-1) can be expressed as power of n, then the product of b and (b-1) could be reduced to a single integer.

If b-1 = cⁿ, then (b-1)bⁿ could be expressed as a single integer in the form dⁿ where d = b * c.

For example,

(17 - 1)17⁴

= (16)17⁴

= (2⁴)17⁴

= (2 * 17)⁴

= 34⁴

Lemma: If integer b in the term bⁿ has factors that can be expressed as power of n, then bⁿ may be expressed as c^z, where, if c > b, then z < n and if c < b, then z > n

Example:

Let bⁿ = 16⁴

16 = 2⁴

Therefore, bⁿ may be written in terms of integer 2 in the form c^z as follows:

16⁴ = 2¹⁶

Rule 6: All A^x where A and x are integers and x>=3, are divisible by A³

Proof:

Aⁿ + (A-1)Aⁿ = A⁽ⁿ⁺¹⁾

If n = 3, then,

A³ + (A-1)A³ = A^{(3 + 1)}

A³ + (A-1)A³ = A⁴

It is seen that the terms A³, (A-1)A³ and A⁴ are divisible by A³.

Again, if n = 4, then,

A⁴ + (A-1)A⁴ = A^{(4 + 1)}

A⁴ + (A-1)A⁴ = A⁵

The terms A⁴, (A-1)A⁴ and A⁵ would be perfectly divisible by A³ as A⁴ is divisible by 3, and the sums of the terms on the left hand side equal to the term on the right hand side would also be divisible by A³.

Rule 7: If integer A in A³ is perfectly divisible by 2, then A³ = 8B³ where B = A/2

Proof:

If A is perfectly divisible by 2, and B = A/2, then, A³ = (2 * B)³

= 2³ * B³

= 8B³

Lemma: To split cubes with even bases: If integer A in A³ is perfectly divisible by 2, then A³ = 8B³ where B = A/2, and A³ is equal to B³ + 7B³

Since A³ = 8B³,

A³ = 8B³ = B³ + 7B³

Rule 8: If integer A in A³ is perfectly divisible by a prime factor P, then A³ = PB³ where B = A/P

Proof:

If A is perfectly divisible by P, and B = A/P, then, A³ = (P * B)³

= P³ * B³

= PB³

Example:

Take the example of 57³

57 is perfectly divisible by prime factor 19.

Therefore, 57³ = 27 x 19³

Lemma: To split cubes with base perfectly divisible by a prime factor: If integer A in A³ is perfectly divisible by a prime factor P, then A³ = PB³ where B = A/P, and A³ is equal to B³ + (B-1)B³

Since A³ = 8B³,

A³ = 8B³ = B³ + 7B³

Rule 9: Fermat's Last Theorem: No three positive integers a, b, c satisfy the equation aⁿ + bⁿ = cⁿ for any integer value of n greater than 2.

Fermat's Last Theorem states it is impossible for a cube to be written as the sum of two cubes, and more generally, no three positive integers a, b, c satisfy the equation aⁿ + bⁿ = cⁿ for any integer value of n greater than 2. An elementary proof for Fermat's Last Theorem is as follows:

Proof:

Pythagoras theorem states:

c² = a² + b²

If we multiply both sides by c,

c³ = ca² + cb²

Since the hypotenuse is greater than the sides, c > a and c > b

Therefore,

ca² > a³ and cb² > b³

=> caⁿ > aⁿ and cbⁿ > bⁿ

=> Fermat's Last Theorem. QED.

In the light of the Repeated Replication Theorem, another proof is also available:

Proof:

It has been proved that,

aⁿ + (a-1)aⁿ = a⁽ⁿ⁺¹⁾

If (a-1) can be expressed as power of n, then the product of a and (a-1) could be reduced to an integer value. If a-1 = bⁿ, then (a-1)aⁿ could be expressed as an integer raised to power n in the form bⁿ

Let (a-1)aⁿ = bⁿ

Then,

aⁿ + bⁿ = a⁽ⁿ⁺¹⁾

which proves Fermat's Last Theorem by contradiction.

Rule 10: Beal's Conjecture: If A^x+B^y=C^z, where A, B, C, x, y, and z are positive integers with x, y, z > 2, then A, B, and C have a common prime factor.

Proof:

It has been shown above that

A^x + (A-1)A^x = A^(x+1)

If (A-1) can be expressed as power of x, then the product of A and (A-1) could be reduced to an integer value. If A-1 = D^x, then (A-1)A^x could be expressed as an integer raised to power y as D^x * A^x = (DA)^x, and therefore as B^x

A^(x+1) may be expressed as an integer raised to power z as C^z

A is either a prime in itself or has prime factors that is common to the three terms. Therefore, A, B and C have a common prime factor.

Rule 11: Repeated substitution of powers in exponential addition

The identity bⁿ + (b-1)bⁿ = b⁽ⁿ⁺¹⁾ may be used to add or subtract numbers that have different exponents or different bases, by repeated substitution of bases and powers. This may be illustrated with the following equations:

Example #1:

2¹⁵ + 2¹⁵ = 2¹⁶

The above equation may be rewritten as follows:

2^{(3 * 5)} + 2^{(5 * 3)} = 2^{(4 * 4)}

8⁴ + 32³ = 16⁴

Example #2:

3³ + 6³ = 3⁵

The exponential identities for 3⁴ and 3⁵ are as follows:

3³ + 2(3³) = 3⁴

3⁴ + 2(3⁴) = 3⁵

Therefore,

(3³ + 2(3³)) + 2(3⁴) = 3⁵

3³ + 2(3³) + 2(3⁴) = 3⁵

3³ + 3³( 2 + 6 ) = 3⁵

3³ + 3³( 8 ) = 3⁵

3³ + 3³( 2³ ) = 3⁵

Thus,

3³ + 6³ = 3⁵

Rule 12: Revised Beal's Conjecture : If A^x+B^y=C^z, where A, B, C, x, y, and z are positive integers with x, y, z > 2, then A, B, and C have at least one common prime factor that is the greatest of the prime factors of A, B and C.

Proof:

It has been shown above that

A^x + (A-1)A^x = A^(x+1)

In the term (A-1)A^x, (A-1) is less than A, if and only if (A-1) could be expressed as an integer with power y, it would be possible to express (A-1)A^x in terms of B^y, in which case, the prime factors of A-1 would be less than A. A⁽ⁿ⁺¹⁾ would not have a prime greater than A, and therefore, either A is prime in itself, or if A has many common prime factors, there would be one which is the greatest among them.

Example #1:

12326391001³ + 28473963212310³ = 12326391001⁴ = 23085737803492332065718589229442591564001

The prime factors of A, B and C are as follows:

A = 12326391001 = 2311*5333791

B = 28473963212310 = 2*3*5*7*11*2311*5333791

C = 23085737803492332065718589229442591564001 = 2311*2311*2311*2311*5333791*5333791*5333791*5333791 = 2311⁴*5333791⁴

The common prime factors of A, B and C are 2311 and 5333791, and 5333791 is the greatest common prime factor of A, B and C.

Example #2:

27081081027001³ + 813244863240840030³ = 27081081027001⁴ = 537853484286584131173623874806259504429852304698108001

A = 27081081027001 = 59*103*509*673*13009

B = 813244863240840030 = 2*3*5*7*11*13*59*103*509*673*13009

C = 537853484286584131173623874806259504429852304698108001 = 59*59*59*59*103*103*103*103*509*509*509*509*673*673*673*673*13009*13009*13009*13009 = 59⁴ * 103⁴ * 509⁴ * 673⁴ * 13009 ⁴

The common prime factors of A, B and C are 59, 103, 509, 673 and 13009, and the greatest common prime factor of A, B and C is 13009.

Example #3:

11103427767506874702903001³ + 2477095567490801422201027824702870³ = 11103427767506874702903001⁴ = 15199464472203083619278077099755610022919025988070927424895999670789670141015099552837081571265612001

A = 1368898397005190663027769554956942040159877633284912763278737738833335709001 = (31*277*317*703763*5796020545213)³

B = 15199464472203083619278075730857213017728362960301372467953959510912036856102336274099342737929903000 = (2*3*5*7*11*13*17*19*23*31*277*317*703763*5796020545213)³

C = 15199464472203083619278077099755610022919025988070927424895999670789670141015099552837081571265612001 = (31*277*317*703763*5796020545213)⁴

The common prime factors of A, B and C are 31, 277, 317, 703763 and 5796020545213, and the greatest common prime factor of A, B and C is 5796020545213.

Conclusion:

A few of the rules relating to addition of integers along with their exponents have been listed above. These rules are easier to view in n-dimensional space and validate its definition² and have practical application. For example, listing the elementary rules relating to addition of integers with exponents enables visualization of the exponential terms in equations. It may be noted that the geometry corresponding to the terms in Beal's Conjecture that necessarily comply with Fermat's Last Theorem could only be as follows:

	Ax	By	Cz
1.	Cube	Cube	Cuboid
2.	Cube	Cuboid	Cube
3.	Cuboid	Cube	Cube
4.	Cuboid	Cuboid	Cuboid

When the terms in Beal's Conjecture are divided by cubes of common prime factors with exponents greater than 3, the remainder left is an integer whose roots indicate the geometric shape of the term. In this context, it is seen that the powers and roots of integers are connected with primality expressing itself in edges of cubes and cuboids in n-dimensional space.

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References:

[¹] https://en.wikipedia.org/wiki/Exponentiation

[²] https://ramanraj.blogspot.com/2017/07/a-clear-definition-of-n-dimensional.html

Symmetrical replication of powers in n-dimensional space

- Ramanraj K

Introduction

Exponentiation is repeated multiplication of the base, and corresponds to symmetrical replication of the base along the axes in n-dimensional space. If b is the base, then its power or exponent n corresponds to dimensional space n extended by symmetrical replication of the base b times along the axes. This addition of exponents in nDspace could be expressed by the following exponential identity:

bⁿ + (b-1)bⁿ = b⁽ⁿ⁺¹⁾

Proof:

bⁿ + (b-1)bⁿ

= bⁿ + b⁽ⁿ⁺¹⁾ - bⁿ

= b⁽ⁿ⁺¹⁾

Illustration of symmetrical replication in n-dimensional space with powers of 2

The figures are rendered with x3dom and can be rotated with pointer.

2 power 1 in 1-D Space

One dimensional space is defined as a stright line along x-axis with a single vertex O in the centre. 2 power 1 would be 2 units along the x-axis.

2¹ = 2

2 power 2 in 2-D Space

Two dimensional space is defined by two intersecting perpendicular straight lines x and y with a single vertex O. The co-ordinates for 2 power 2 are 2 units along x-axis and y-axis, making a square with 4 square units.

2² = 2 x 2 = 4

2 power 3 in 3-D Space

Three dimensional space is defined by two intersecting perpendicular straight lines x and y with a single vertex O, and a third straight line z perpendicular to both x and y passing through O. The co-ordinates for 2 power 3 are 2 units along x-axis, y-axis and z-axis, making a cube with 8 subcube units.

2³ = 2 x 2 x 2 = 8

2 power 4 in 4-D Space

Four dimensional space is the 3-D space defined by lines x, y and z as above whose vertex O₁ lies on a straight line w with vertex O. Therefore, in 4-D space at least two vertices co-exist and the distance between the vertices O and O₁ is defined by the 4th dimension w. The lines x and w coincide. The co-ordinates for 2 power 4 are 2 units along x-axis, y-axis, z-axis and w-axis making two cubes with 8 subcube units, and a total of 16 subcube units.

2⁴ = 2 x 2 x 2 x 2 = 16

It may be noted that the exponential identity for addition of exponents in n dimensional space is

bⁿ + (b-1)bⁿ = b⁽ⁿ⁺¹⁾.

2⁴ involves addition of two exponents as follows:

2³ + (2-1)2³ = 2⁽³⁺¹⁾

2³ + 2³ = 2⁴

This addition of 2³ and 2³ results in symmetrical replication of cubes (b-1) times extending to the next dimension resulting in 2⁴ corresponding to the fourth dimension. In the fourth dimension along the w-axis, 2³ with 8 subcube units is replicated the base times. When seen from the third dimension, it would be 16 subcube units.

2 power 5 in 5-D Space

Five dimensional space is the 3-D space defined above whose vertex lies in a 2-D plane. Therefore, in 5-D space at least two vertices exist and the distance between them is defined by the 4th and 5th dimensions. The co-ordinates for 2 power 5 are 2 units along x-axis, y-axis, z-axis, w-axis and v-axis making four cubes with 32 sub cube units.

2⁵ = 2 x 2 x 2 x 2 x 2 = 32

The addition of 2⁴ and 2⁴ again results in symmetrical replication base times along v-axis extending to the next dimension resulting in 2⁵ corresponding to the fifth dimension. In the fifth dimension along the v-axis, 2⁴ is replicated the base times resulting in a cuboid with 4 two cube units. When seen from the third dimension, it would be 32 subcube units.

2 power 6 in 6-D Space

Six dimensional space is the 3-D space whose vertex lies in another 3-D space. In 6-D space at least two vertices exist and the distance between the two is defined by the 4th, 5th and 6th dimensions. The co-ordinates for 2 power 6 are 2 units along x-axis, y-axis, z-axis, w-axis, v-axis and u-axis making eight cubes with 64 sub cube units.

2⁶ = 2 x 2 x 2 x 2 x 2 x 2 = 64

The addition of 2⁵ and 2⁵ again results in symmetrical replication base times along v-axis extending to the next dimension resulting in 2⁶ corresponding to the sixth dimension. In the sixth dimension along the u-axis, 2⁵ is replicated the base times resulting in a cube with 8 subcube units when viewed from the sixth dimension and 64 subcube units from 3D point of view.

This may be repeated n power times in n dimensional space.

Illustration of symmetrical replication in n-dimensional space with powers of 3

3 power 3 in 3-D Space

The co-ordinates for 3 power 3 are 3 units along x-axis, y-axis and z-axis, making a cube with 27 subcube units.

3³ = 3 x 3 x 3 = 27

3 power 4 in 4-D Space

The co-ordinates for 3 power 4 are 9 units along x-axis and 3 units along y-axis and z-axis in 3-D space. In 4-D space, the 3x3x3 cube in 3-D space is symmetrically replicated three times along w-axis resulting in a cuboid with 81 subcube units.

3⁴ = 3 x 3 x 3 x 3 = 81

The exponential addition for 3⁴ using the identity bⁿ + (b-1)bⁿ = b⁽ⁿ⁺¹⁾ is as follows:

3³ + (3 - 1)3³

=3³ + (2)3³

=27 + (2)27

=27 + 54

=81

=3⁴

3 power 5 in 5-D Space

The co-ordinates for 3 power 5 are 9 units along x-axis and 9 units along y-axis and 3 units along z-axis in 3-D space. The three 3x3x3 cubes in 4-D are symmetrically replicated three times along v-axis resulting in a cuboid with 81 subcube units.

3⁵ = 3 x 3 x 3 x 3 x 3= 243

The exponential addition for 3⁵ using the identity bⁿ + (b-1)bⁿ = b⁽ⁿ⁺¹⁾ is as follows:

3⁴ + (3 - 1)3⁴

=3⁴ + (2)3⁴

=81 + (2)81

=81 + 162

=243

=3⁵

3 power 6 in 6-D Space

The co-ordinates for 3 power 6 are 9 units along x-axis, y-axis and z-axis in 3-D space. In 6-D space, the 3⁵ cuboid in 5-D space is symmetrically replicated three times along u-axis resulting in a cube with 729 subcube units.

3⁶ = 3 x 3 x 3 x 3 x 3 x 3 = 729

The exponential addition for 3⁶ using the identity bⁿ + (b-1)bⁿ = b⁽ⁿ⁺¹⁾ is as follows:

3⁵ + (3 - 1)3⁵

=3⁵ + (2)3⁵

= 243 + (2)243

= 243 + 486

= 729

=3⁶

Repeated substitution of powers in exponential addition

The identity bⁿ + (b-1)bⁿ = b⁽ⁿ⁺¹⁾ may be used to add or subtract numbers that have different exponents or different bases, by repeated substitution of bases and powers. This may be illustrated with the equation 3³ + 6³ = 3⁵

The exponential identities for 3⁴ and 3⁵ are as follows:

3³ + 2(3³) = 3⁴

3⁴ + 2(3⁴) = 3⁵

Therefore,

(3³ + 2(3³)) + 2(3⁴) = 3⁵

3³ + 2(3³) + 2(3⁴) = 3⁵

3³ + 3³( 2 + 6 ) = 3⁵

3³ + 3³( 8 ) = 3⁵

3³ + 3³( 2³ ) = 3⁵

Thus,

3³ + 6³ = 3⁵

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