Existential Number Theory What is an existential number?
Existential Number Theory
What is an existential number?
2024/08/07
Haruo Nagano
Existential number theory is based on the concept that
"this thing is this thing."
With this in mind,existential numbers should be considered.
If "this thing is not this thing." then there would be no need
for people to worry or think about the universe.
Because the universe is not the universe.
We must be aware that there are two worlds in mathematics (the
world of existential numbers and the world of ultimate
numbers) and use that as a basis to pursue the essence of the
universe.
The universe itself is a world of existential numbers.
Define 'existential number.'
An existential number is "two natural numbers and the
relationship between them."
The mathematical origins of this definition have been
explained elsewhere and will not be explained here.
As an example of existential numbers,
２＋３ , ５－４ , ６×７ , ８÷９
Anything that is not an existential number is called an
ultimate number.
For example,
２, ３.４, －５, ６.７７７・・・, √－８,
The following is also not an existential number,
０.1＋２, ４－√５, ６.３３３・・・×３, ８÷√９
The following equation is true.
But, it does not hold true as an equation of existential
numbers.
２＋３ ＝ ７－２
Since the relationship between the two numbers is different on
both sides, we need to make them the same.
Therefore, if we replace the right-hand side as follows, we
get the equation for existential numbers.
２＋３ ＝（４＋３）－２
２＋３ ＝ ４＋（３－２）
Therefore,
２＋３ ＝ ４＋１
If this is true for any equation, then it can surely be
rewritten as an equation for existential numbers.
"If the equation holds, then the equation of existential
numbers must hold."
For example
０.1＋２.９ ＝ ３
This equation is valid as a calculation, but not as an
equation of existential numbers.
However, this equation can be rewritten as an equation of
existential numbers.
By replacing the 3 on the right-hand side with 1 + 2 and
multiplying both sides by 10, an equation of existential
numbers is created, which holds true.
０.1＋２.９ ＝ １＋２
１＋２９ ＝ １０＋２０
If the equation holds, then an equation for existential
numbers can be created and holds.
This can be used in a variety of problems dealing with natural
numbers.
For example, it can be applied to problems such as the
existence of a general solution to equations, Fermat's Last
Theorem, the abc conjecture, and the Riemann hypothesis.
As an application :
Proof that there is no general solution to equations of degree
5 or higher (non-exact proof)
（Ｘ＋Ａ）^２＝Ｂ
（Ｘ＋Ａ）^３＝ＢＸ＋Ｃ
（Ｘ＋Ａ）^４＝ＢＸ^２＋ＣＸ＋Ｄ
（Ｘ＋Ａ）^５＝ＢＸ^３＋ＣＸ^２＋ＤＸ＋Ｅ
（Ｘ＋Ａ）^６＝ＢＸ^４＋ＣＸ^３＋ＤＸ^２＋ＥＸ＋Ｆ
．．．．．．．．．．
If a general solution exists, then X must exist for any
natural numbers Ａ，Ｂ，Ｃ，Ｄ，Ｅ，・・・.
That is, their number is not restricted.
There exist A, B, C, D, E, F,・・・ , such that the value of X
is a natural number a.
With this in mind, consider the equation for existential
numbers below.
Proof of a cubic equation
（Ｘ＋Ａ）^３＝ＢＸ＋Ｃ＝（Ｂ＋Ｄ）Ｘ＝ＥＸ
From Ｒ｜ＤＸ ⇒ Ｒ｜Ｃ
At Ｘ＝a
Ｒ^３｜（a＋Ａ）^３ ⇔ Ｒ^３｜Ｅa （ Ｅ＞a２ ）
Proof of the quartic equation
（Ｘ＋Ａ）^４ ＝Ｂ（Ｘ＋Ｃ）^２＋Ｄ
＝Ｂ（Ｘ＋Ｃ）^２＋Ｅ（Ｘ＋Ｃ）^２
＝Ｆ^２（Ｘ＋Ｃ）^２
Ｒ^２｜Ｅ（Ｘ＋Ｃ）^２⇒ Ｒ^２｜Ｄ
From Ｒ^２｜Ｆ^２⇔ Ｒ^２｜Ｂ＋Ｅ
At Ｘ＝a
Ｒ^２｜（a＋Ａ）^２ ⇔ Ｒ^２｜Ｆ（a＋Ｃ）
Proof of a quintic equation
（Ｘ＋Ａ）^５＝Ｂ（Ｘ＋Ｃ）^３＋ＤＸ＋Ｅ＝Ｆ^３（Ｘ＋Ｃ）^３
Ｒ^３｜Ｇ（Ｘ＋Ｃ）^３ ⇒ Ｒ^３｜ＤＸ＋Ｅ
Ｒ^３｜Ｆ^３ ⇔ Ｒ^３｜Ｂ＋Ｇ
Ｒ^５｜（Ｘ＋Ａ）^５ ⇔ Ｒ^５｜Ｆ^３（Ｘ＋Ｃ）^３
The equation of existential numbers does not hold.
For equations of degree 5 , the equation of existential
numbers does not hold, so there is no general solution for
equations of degree 5 .
Equations of degree 6 or higher do not hold for some reason.
Supplement for those who have difficulty understanding :
Ｒ^ｎ｜Ａ is the nth power of the natural number Ａ,Ｂ^ｎ.
When Ｒ｜Ａ ＝ Ｒ^ｎ｜Ｂ^ｎ holds, Ａ is said to be
restricted to Ａ＝Ｂ^ｎ.
Under this restriction, the equation of existential numbers
holds.
Ｒ^ｎ｜Ａ ⇔Ｒ^ｎ｜Ｂ^ｎ
Also, since Ｒ^ｎ｜Ｂ^ｎ ⇔Ｒ｜Ｂ^ｎ
Ｒ^ｎ｜Ａ ⇔Ｒ｜Ｂ^ｎ
At ａ^２＝ｂ
If Ｒ｜ａ^２ ＝ Ｒ｜ｂ holds, then b is restricted by ａ^２,
but if Ｒ^２｜ａ^２ ＝ Ｒ^２｜ｂ then b is not restricted.
Ｒ^２｜ａ^２ ⇔ Ｒ^２｜ｂ
For Ｒ^２｜Ｒ^３｜Ａ^３ ＝ Ｒ^３｜Ｒ^２｜Ｂ^２
Ｒ^２｜Ａ^３ ⇔ Ｒ^３｜Ｂ^２ .
If Ａ ＝ Ｘ^２ , Ｂ ＝ Ｙ^３ , then
Ｒ^６｜Ｘ^６ ⇔ Ｒ^６｜Ｙ^６ or
Ｒ｜Ｘ ⇔ Ｒ｜Ｙ .
Since the equation of existential numbers holds, the equation
Ｒ^２｜Ｒ^３｜Ａ^３ ＝ Ｒ^３｜Ｒ^２｜Ｂ^２ holds.
If a general solution to the equation exists, then the
equality of existential numbers must hold even if Ａ，Ｂ，
Ｃ，Ｄ，Ｅ，．．are not restricted.
For example, about a cubic equation
When Ｘ＝２,
Ｒ^３｜（２＋Ａ）^３ ⇔ Ｒ^３｜２Ｅ
Ａ is restricted to ２Ｆ (but E ≧ 32),
but this restriction is avoided by setting X = 1.
For the fourth-order equation, The restriction is avoided by
setting X = 1. (but E ≧ 27)
For quintic equations (and above), restriction are
unavoidable.
Ｒ^２ｎ｜Ａ^２ｎ ＝ Ｒ^２ｎ｜Ｂ^ｎ
This equation holds true as an equation of existential
numbers,
but
Ｒ^ｎ＋４｜Ａ^ｎ＋４ ＝ Ｒ^ｎ＋４｜Ｂ^ｎ＋２
This equation does not hold as an equation of existential
numbers.
There are two worlds in mathematics.
To know the true nature of the universe, one must
know the world of existential numbers, not the world
of ultimate numbers.
Because the universe is existential numbers itself.
Throw away the'prejudice'
Be 'skeptical' in everything
Have a strong 'inquiry spirit'
Pay close attention to
Then you will surely find the correct answer
Advice from the author
"What you don't know is neither shame nor
incompetence.
It is a shame and incompetence not to make an effort
to know. "
From the author
「 Ask, and it shall be given you ;
seek, and ye shall find ;
knock, and it shall be opened unto you ; 」
Matthew 7:7
五
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