Friday, May 01, 2020

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

- Ramanraj K

1st 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 valueGeometric space
111 Length
121 x 1 Area
131 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 valueGeometric space
1111 Length
1211 x 11 Area
1311 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 valueGeometric space
1313 Volume
1311 x 12 Volume = Height x Area
1311 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 cz, 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 cz as follows:

164 = 216

Rule 6: All Ax 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 Ax+By=Cz, 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

Ax + (A-1)Ax = 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 = Dx, then (A-1)Ax could be expressed as an integer raised to power y as Dx * Ax = (DA)x, and therefore as Bx

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

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 Ax+By=Cz, 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

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

In the term (A-1)Ax, (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)Ax in terms of By, 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:

AxByCz
1.CubeCubeCuboid
2.CubeCuboidCube
3.CuboidCubeCube
4.CuboidCuboidCuboid

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.


References:

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

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

Tuesday, April 14, 2020

Symmetrical replication of powers in n-dimensional space

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