Science,English,and math Ⅲ

【Involution increases explosively】

〈A calculation made Hideyoshi panic〉

Hideyoshi Toyotomi who was a military commander in the Warring States period was a sharp person, but wasn`t good at reading and writing very much, so there were his plenty of vassals called お伽衆 who talked him about learnings and their experiences.

While one of them, Sinzaemon Sorori was a good artisan who was good at making sheath, it is said he was the founder of a comic storyteller, so there are lots of his episodes which were quick witted.

One day Hideyoshi praised Shinzaemon and tried to reward him. What do you...

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【Involution increases explosively】

〈A calculation which made Hideyosi panic〉

What do you want? Hideyoshi asked Shinzaemon. Shinzaemon thought it of for a while and answered.

First day, a grain of rice, second day, two grains of rice, third day four grains of rice, and fourth day eight grains of rice. Thus starting from a grain of rice, please give me grains of rice which is two times of the previous day for a month.

Then Hideyoshi promised to do it too readily, but with each passing days, he was in difficulty due to his promise with Shinzaemon.

If he gives Shinzaemon grains of rice...

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【Involution increases explosively】

〈A calculation which made Hideyoshi panic〉

If Hideyoshi gives Shinzaemon grains of rice as he was asked, it`s just about 1 go which means 180 ml in two weeks. To be exact it`s 8192 grains. 1 go is about 6500 grains.

However it reaches about a half billion and thirty million grains in a month. It`s equal to 200 bales. 1 bale is about 12 kilogram, so 200 bales are 12 ton. It`s an exorbitnat number.

Hideyoshi was aware of it before a month passed, and made Shinzaemon change his reward in a haste.

〈If folding a newspaper forty two times....〉

Multiplying...

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【Involution increases explosively】

〈If folding a newspaper forty two times...〉

Multiplying the same number repeatedly like 2×2×2 is called an involution.

If going on muliplying the same number, the involution increases explosively in its process.

For example, when folding a newspaper, let`s calculate its thickness. If the thickness of the newspaper is 0.1 millimeter, when folding n times, its thickness is 0.1×2 to the nth power.

When folding ten thimes, its thickness is about 10 centimeters, and 14 times about 164 centimeters which is a little taller than a grown-up of a female, but...

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【Involution increases explosively】

〈If folding a newspaper forty two times...〉

...but after that it increases sharply.

When thirty times, it reaches about 107 kilometers which is distance from Tokyo to Atami, and when forty two times, to my surprise, it`s over 0.38 million kilometers which is distance between the earth and the moon.

Though we can`t fold it actually, we can grasp an image in which the involution increases explosively, can`t we?

〈Simple interest is different from compound one so much〉

Expansion of multiplying the same number is an exponential function which we learn in...

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【Involution increases explosively】

〈Simple interest is different from compound one so much〉

Expansion of multiplying the same number repeatedly is exponential function which we learn in high school. It is said the exponential function is one of the closest function in our life, especially the compound interest method is so close to us that it occured to us at onece.

The compound interest method is adding an interest in a fixed period to a principle and it`s the second principle, and adding the interest in the fixed period to the second principle, and it`s the third principle, and adding....

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【Involution increases explosively】

〈Simple interest is different from compound one so much〉

Thus adding the interest to the principle, and adding the interest to the new principle, it reapeats. It`s the compound interest method.

On the other hand, the simple interest method is without adding the interest of the principle in the last time, calculating the interest to the principle alone.

Let`s suppose that the principle we deposited was a million yen and its interest was ten % a year, its total of the principle and interest is a million and a hundred thousand yen both in simple and ...

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【Involution increases explosively】

〈Simple interest and compound one is different so much〉

... so the total of the principle and interest is a million and a hundred thousand yen in both the simple and compound interest in the first year, but after the next year is different.

As to the compound interest method, adding the ten % of interest to a million and a hundred thousand yen which is the total of the principle and interest after a year, so its total in relation to after two years is 110＋110×10%＝121, so it`s a million and two hundred ten thousand yen.

On the other hand, the simple...

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【Involution increases explosively】

〈Simple interest and compound one is different so much〉

On the other hand, as to the simple interest method is as adding the interest to the first principle alone,110＋100×10%＝120.

Two of them is differen by only ten thousand yen, but when continuing several years difference between the two is obvious, for after a decade, its difference is six hundred thousand yen.

While the compound interest is an involution which means mulitiplying 1.1 by 100, the simple one is adding a hundred thousand to the principle repeatedly alone.

Though it`s a period of low....

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【Involution increases explosively】

〈Simple interest and compound one is different so much〉

As it`s the time of low interest extremely at present, even if the interest to a bank saving is high, it`s at best 0.3% a year. It`s a time deposit in the net bank.

When depositing a million yen in the bank for a decade, both of the simple interes and compound one isn`t different so much. It`s difference is only four hundred and eight yen.

〈Population increases like a geometric series〉

If using the exponetial function which is the expansion of the involution, except for caluculation of the ...

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【Involution increases explosively】

〈Population increases like a geometric series〉

If using the exponetial function which is the expansion of the involution, except for the caluculation of the compound interst, we can describe lots of social and natural phenomena.

Malthus who wasn`t only a priest but an English economist who played an active part in Europe from end of the 18th to the beginnig of the 19th century said in his work, Mathematician theory of population like tne next.

From now on while the population increases like a geometric series, the food does like an arithmetic series,so...

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【Involution increases explosively】

〈Population inceeases like geometric series〉

...so we would suffer from food crisis in the future.

The way of increasing like the geometric ...Oh, I made a mistake again. It`s not the geometric series but geometric progression, and it`s not arithmetic series but arithmetic progression.

The way of increasing like the geometric progression is continuing multiplying the first number by the same number, for example, 1,3,9.27....

It goes on multiplying the first number by 3. The way of increasing is the involution, then the population is showed with the...

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【Involution increases explosively】

〈Population increases like a geometric progression〉

The way of increasing is involution, and it`s showed with an exponential function.

On the other hand, a land where we grow crops, or resource which we keep a domestic animal is limited, so the food doesn`t increase exponentially.

Even if going on increasing ideally, it would do like the arithmetic progression, adding the first number to the same number, 1.4.7.10... Malthus thought like that.

Then the amount of the food is showed with a straight line, a liner function.

Actually when looking at a...

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【Involution increases explosively】

〈Population increases like a geometric progression〉

Actually when looking at a change on population in the world, it`s increases such a high speed extremely that we can call it explosion on population.

The population used to be no more than about a billion in 1800s, but it reached a billion and six hundred million after a hudred years. Two billion and five hundred million in 1950, six billion and a hundred million 2000, seven billion and three hundred million 2015. Some people estimate the population will reach ten billion in 2065.

On the other hand....

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【Involution increases explosively】

〈Population increases like a geometric progression〉

On the other hand, the population goes on decreasing after 2007 in Japan, for the birthrate which means average number of a female is given a birth in her lifetime becmes low.

Actually it was 4.54 in 1947, but was 1.25 in 2005. It becomes high to 1.4 at present, but it doen`t reach 2.07 which is a birthrate for maintaining the population.

Without being born two children from parents, the population would decrease, so the birthrate which maintains the population is about 2.

〈How much is the distance ...〉

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【Involution increases explosively】

〈How much is the distance where a soup doesn`t become cool?〉

According to a law on cooling by Newton, a degree of when a thing loses its fever and becomes cool is in proportion to difference on temperature between a thing and its surrounding.

For example when a miso soup of which temperature is 80 degree becomes cool in a room of which temperature is 20 degree, the difference between the temperatrue is 60 degree at first, so the degree of becoming cool in fixed time is big.

Let`s suppose that it became 45 degree after 15 minutes. Then the difference...

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【Involution increases explosively】

〈How much is distance where a miso soup doesn`t become cold?〉

Then the difference between the thing and the room is 40, so the degree of becoming cool becomes smaller than the first one.

As time passed, the miso soup became 25 degree then the differece between the thing and the room is no more than 5 degree, so the way of becoming cold is very slowly.

In short the way of becoming cold on the miso soup is sharply at the start, but is mild gradually. Actually that change is expressed with the exponential function.

By the way I`m afraid I may have...

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【Involution increases explosively】

〈How much is distance where a miso soup doesn`t become cold?〉

By the way I`m afraid I may have drifted away from the subject but, some young generations learn to be independent from their parents and to live from their parents separately.

Then we sometimes call its appropriate distacnce between the two generation the one where a soup doesn`t become cold. How much is the distance actually?

The way of becoming cold on liquid is greatly influenced by the temperature of the outside, its total area, conductivity on the heat of a vessel, so it`s hard to say...

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【Involution increases explosively】

〈How much is distance where a soup doesn`t become cold?〉

...but a researcher annonced like the next.

The miso soup in a pan made from stainless takes a half hour from 90 degree which just after cooking to 65 degree which is the best time to eat.

So the distance where the soup doesn`t become cool is the one of an half minute on foot, which means about two thousand a hundred meters.

It originally means that when something bad happens to their old parents, the distance where they can rush to at once, but both of a generation of working couple and the...

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【Involution increases explosively】

〈How much is the distacne where a soup doesn`t become cool?〉

...but as both of a generation of working couple and the healthy old age have increased in the present day, its meaning seems to change into a distance the old age can support the child care.

In general,when a changeable ration on a variable in an instant is proportion to the value of the variable, the variable increases or decreases according to the exponential function.

If trying to express it with an equation, it`s like the next, we can skip it, the author said, I`ve tried to understand...

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【Involution increases explosively】

〈How much is distance where a soup doesn`t become cool?〉

...so I`m going to leave it out.

There are lots of phenomena which cause a sharp change by the effect of involution around us, and we`re frequently surprised at the degree of change.

However it`s expressed with the exponential function which isn`t complete but an elementary function, so I`m impressed with it, the author said like that.

Actually it was learned by a student of humanity in high school as well. Moreover it isn`t limited in this time alone.

Every time natural phenomenon which had...

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【Involution increases explosively】

〈How much is the distance where a soup doesn`t become cool?〉

Whenever natural phenomena which had existed before people invented math or works of social activities by free will of people were expressed with the math, they`ve made the author realize keenly the math`s power and interest, and he`s felt like believing its vast possiblility, he said like that.

Speaking of high school, it`s about forty years ago for me, so I can`t remember it at all.

【A queen in math and a miracle in integral numbers】

〈Shaking a dice three times and creating a number of...〉

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【A queen in math and miracles in integral numbers】

〈Shaking a dice three times and creating a number of three digit〉

If there is a game like the next, will you join it?

Shaking a dice three times and combinating them at will and creating a number of three digit. When each number is 1,5,6, we can create the number of 156 or 561.

Then writing down the number of three digit twice repeatedly, if it`s 156, 156156, after that dividing the number of six digit by seven and if there is a remainder its your lucky number.

Then you can receive a bill of ten thousand yen which is the same number of...

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【A queen in math and miracles in integral numbers】

〈Shaking a dice three times and and creating a number of three digit〉

You can receive a bill of ten thousand yen which is the same number of your lucky number. Its joining expenses is 1000 yen.

The lucky number is the remainder divided by 7, so it`s either 0,1,2,3,4,5,6. Highest prize is sixty thousand yen.

Unless you are so unlucky that your lucky number is zero, there may have been lots of people who want to join it, for they expect to win at least ten thousand yen.

Just a moment! I don`t recommend you join the game, the author said...

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【A queen in math and miracles in integral numbers】

〈Shaking a dice three times and creating a number of three digit〉

For example, let`s figure in the case of 156156 actually.

Dividing 156156 by 7 is equal 22308,it`s divisible, so there is neither remainder nor lucky number, but it`s not accidental.

Whneever you join the game, its prize is always zero.

The author said he was going to reveal its secret of the magic tric.

Creating the number of three digit twice repeatedly is as same as multiplying the number of the three digit by 1001, and 1001 is divided by seven, so your lucky is...

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【Queen in math and miracles in integral numbers】

〈Shaking a dice three times and creating a three digit number〉

...and your lucky number is always zero.

By the way, using the dice is emphasizing a character on game, so whatever the three digit number written repeatedly is all right.

Is it right in the world? I`ve tried line up some number from one to six at random and to divide it by seven a few times, it`s divided, but I can`t try all the combinations from one to six. I`m sure I would be tired of it.

I`ve wanted to make sure whether or not the book says is right, but I find it tiresome.

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【Queen in math and miracles in ingeral numbers】

〈A memo left by a great mathematician Fermat〉

Zero and increasing or decreasing from zero one by one, we can get some numbers. We call all of them integral numbers, and we call a mathematical field in which we study on the integral numbers a number theroy.

While the integral numbers have been familiar with us, lots of their proper have been covered with something mysterious.

For example, when n is an integral number which is more than three, there are no natural numbers which meet the next equation, x to the nth power ＋ y to the nth power＝...

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【Queen in math and miracles in integral numbers】

〈A memo left by a great mathematician, Fermat〉

For example, if n is an integral number more than three, there are no natural numbers which meet the next equitation, x to the nth power ＋ y to the nth power ＝ Z to the nth power, x,y, or z.

We call a law by Fermat.

A French mathematician who played an active part in the 17th century left a memo in the blnak of a book like the next.

I`ve found a splendid genuine proof on the theorem, but it`s too long to write down on the blank, but almost all of the mathematicians at present have thought...

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【Queen in math and miracles in integral numbers】

〈A memo left by a great mathematician, Fermat〉

...but almost all of the mathematicians at present have thought the way of proving by Fermat had a mistake or insufficiency, for it was in 1994 more than after three hundred years Fermat was dead when it was demonstrated by an English mathematician actually.

It`s way is complicated one in which technical skill in modern math has been made full use of.

〈Perfect number is limited〉

There are various characters in integral numbers, and each of them has a name, for example, natural numbers....

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【Queen in math and miracles in integral number】

〈Perfect numbers are limited〉

...for example, natural numbers, prime numbers, even numbers, odd numbers, triangular numbers, square numbers, amicable numbers, Pythagoras numbers and so on, and one of them is perfect numbers which look nice. Have you ever heard of it? I haven`t.

When an integral number a is divided by b, b is a divisor belonging to a.

In a range of plus integral numbers, except for itself, when adding all divisors is equal to the original number, we call it a perfect number.

The smallest perfect number is six. The perfect...

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【Queen in math and miracles in integral numbers】

〈Perfect numbers are limited〉

Perect numbers continue 6, 28, 496, 8128...but there are four of them alone below ten thousand. Number of 51 of perfect numbers have been found until now.

Fifty first perfect number which was found in 2018 was more than fourty nine million digit, and is a huge number. The study on the perfect numbers has gone on since A.D.4th century, but 51 alone was found, so they are rare, but it seems that mathematicians expect there are numberless perfect numbers though it has never been proved.

This is just an aside,...

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【Queen in math and miracles in integral numbers】

〈Perfect numbers are limited〉

This is just an aside, it is said what the perfect number is six is related the God created the world in six days. The seventh day is the one for taking a rest, a Sunday.

Saint Augustine of the first archbishop of Canterbury who was well known of propagation for the Christianity said like the next.

Six itself is a perfect number. It doesn`t always mean that the God created everything in six days. It means opposite.

Moreover six is multiplying the first prime number, 2 by 3, so its multiple is divided by...

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【Queen in math and miracles in integral numbers】

〈Perfect numbers are limited〉

...so the multiple is a useful number which is frequently divided by various numbers. Actually lots of numbers around us, for instance, twelve months, twenty four hours, thirty days, sixty minutes, and three hundred sixty degrees, are multiples of six.

By the way, as to 28 which is the perfect number next to six, it`s a total number of adding protons and neutrons which is stable for atomic nucleus, or is coincided with number of bone which made up with skull, except for hyoid bone 舌骨, or with the number of....

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【Queen in math and miracles in integral numbers】

〈Perfect numbers are limited〉

... or is coincided with the number of teeth on a grownup, except for wisdom tooth.

Moreover after 28 years, as repeating a leap year seven times,a day of month and the day of week makes around once, so even if twenty eight years passed, we can use the calendar of twenty years ago as it is.

I`m going to express several integral numbers simply.

Natural number is a positive integral number. We have to be careful that zero doesn`t included.

Prime number is an integral number which is bigger than two and it`s...

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【Queen in math and miracles in integral numbers】

〈Perfect numbers are limited〉

A prime number is an integral number which is bigger than two and is divided by one and itself alone, for example, 2,3,5,7,11,13,17,19.... I`m going to express on prime number in details later.

Even number is an integral number which is divided by two.

Odd number is an integral number which isn`t divided by two.

Triangular number is a total number which are arranged with a equilateral triangular 正三角形　way. For example, 1,3,6,10...

Square number is an integral number which is a square of a natural numbe...

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【Queen in math and miracles in integral numbers】

〈Perfect numbers are limited〉

Squared numbers are squared of natural numbers, like 1,4,9,16,25,36,49,64.....

Amicable numbers are a conbination of two natural numbers. Except for itself,a sum of divisors is equal to the other number.

For example, sum of divisors on 220 is 284, and the other sum of divisors on 284 is 220, and the two of them are amicable numbers.

I`m sorry, my explanation is hard to understand, I`m afraid.

The number of Pithagoras is a combination of three integral numbers which are three sides on a right triangle.

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【A queen in math and miracles in integral numbers】

〈Perfect numbers are limited〉

For example,(3.4.5)(5.12.13) are the number of Pithagoras.

〈A mystery in 6174〉

I`m going to express one more mysterious proper on integral numbers. It`s the one in the number of 6174.

The author said he wrote the draft on this book in 2019, and thinking on the 2019.

Using the four number in the four digit, and making the biggest number and the smallest number, the biggest one is 9210 and the smallest one is 0129, it means 129.

What is the remainder when you subtract 129 from 9210? It`s 9081, and doing...

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【A queen in math and miracles in integral numbers】

〈A mystery in 6174〉

...and doing the same thing on 9081, and it`s 9621, and repeating on 9621, it`s 8352. As to the 8352, the biggest one is 8532 and the smallest one is 2358, so it`s 6174.

It`s not interesting in particular until then, and I`m afraid it`s boring, but if repeating the same thing whatever we choose the four digit numbers at first, it will reach 6174 finally. You will be surprised at it very much, won`t you?

Everyone can make sure it easily, so please try it with the year when you were born, the author said like that.

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【A queen in math and miracles in integral number】

〈A mystery in 6174〉

I`ve tried it with the year when I was born, to my surprise,it`s 6174.

However when the samd number is used for the four-digit, like 9999 it becomes zero.

We call the number which has the proper like that Kaprekar number. It`s the name of an Indian math who discovered the proper.

The Kaprekar number on the four-digit is 6174 alone, and on the three-digit is 495, and on the six-digit is 549945 and 631764.

As to the five-digit, there is no the Kaprekar one. Number of 20 Kaprekar number have been discovered until now.

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【A queen in math and miracles in integral numbers】

〈A mystery in 6174〉

One of the greatest mathematicians in the 19th century, Gauss said the number theory is a queen in math.

The author said he`s thought the number theory isn`t only the hardest to understand but lots of ideas on the number theory are beautiful.

In addition the techinical skill or theory on the number theory are so unique that they have rarely been put to practical use in other field. It keeps itself above all the vulgarity around it, so the number theory seem to be as if it were a queen in math, though I`m not sure of it.

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【Prime numbers have never been solved】

〈What is the most important number?〉

What is a character on your birthday? If it`s July 16, 16 is an even number, a multiple of four, two to the fourth power, and four squared.

On the other hand how about the seven?

We frequently call it the lucky seven, so it seems to be auspicious, but we apt to think it doesn`t have any conspicuous proper, but it has a proper that it isn`t divided, except for 1 and itself.

We call an integral number which has the proper like that and is bigger than 1 is a prime number.

Except for the prime number, all numbers...

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【Prime numbers have never been solved】

〈What is the most important number?〉

Except for the prime numbers, all the numbers are products of multipled by the prime numbers like 6 ＝2×3. It`s a prime factorization, and the prime number is the prime one in the numbers literally, and the most important one.

So it is no exaggeration to say that the prime number is the most important one in all the numbers.

While it`s very important, if looking for it from a small one, it looks to be at random.

The study on the prime number has started since two thousand years ago at the time of Ancient Greece.

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【The prime numbers have never been solved】

〈What is the most important number?〉

The study on the prime number has been done at present, and lots of mathematicians are interested in whether or not the way of appearing on the prime number has a rule.

〈A proof which has been offered of a million dollars〉

As to a proof on a distribution of the prime numbers, Riemann hypothesis is well known. It was advocated by a German mathematician, Berhard Riemann in 1895.

Its expactation on the concrete content of the Riemann hypothesis is very hard to understand, so I`m going to omit, the author said.

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【The prime number have never been solved】

〈A proof which has been offered a reward of a million dollars〉

However, if the Riemann hypothesis is right, it means that all the prime numbers which look at random at first sight have a common order.

However the proof on Riemann hypothesis has never been solved in 2019, and a reward of a million dollars has been offered by an Ameircan research institute.

There is a law on the prime number which has never been demonstrated it`s right, though its exception has never been found like the Riemann hypothesis.

All the even numbers which are bigger...

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【The prime numbers have never been solved】

〈The proof which a million dollars of reward has been offered〉

It`s all the even numbers which are bigger than three are expressed with addition by two prime numbers.

For example, 4 ＝ 2＋2, 6 ＝ 3＋3, 8 ＝ 3＋5, 10 ＝ 3＋7, 12 ＝ 5＋7, 14 ＝ 3＋ 11, 16 ＝ 3＋ 13, 18 ＝ 5＋13.....

As to by far bigger even numbers, it`s the same. Please try with a varied even numbers.

It has been proved that all the even numbers which are bigger than three have been shown with addition of the prime numbers until four hundred 京, the 京 is ten thousandth times as big as a trillion.

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【Prime numbers have never been solved】

〈A proof which a reward of a million dallars has been offered〉

It has been proposed by Christian Goldbach in Prussia in 18th century, so they call it Goldbach hypothesis, and no one has proved it`s right until at present, contrary to it, no one has proved it`s in the wrong until at present.

Prussia used to be northeast to modern German.

And when two successive odd numbers like 11, 13 are prime numbers, we call them twin prime numbers, and it has been unclear whether or not the twin prime numbers exist infinitively.

In spite that prime numbers are...

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【Prime numbers have never been proved】

〈A proof which a reward of a million dollars has been offered〉

In spite that the pirme numbers are prime for all the numbers, their proper has been covered in darkness.

A German mathematician said the God has created the integral numbers , but the author said it`s the prime numbers which the God has created, and he`s thought the God`s enjoyed a riddle for all the intellects in the universe through the prime numbers.

Possibly the one which may hasve succeeded in solving the riddle faster than the human being somewhere in the huge universe.

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【Prime numbers have never been solved】

〈Is today a day of prime numbers?〉

The author said it was eleventh day August in 2019 when he wrote the draft of the book which I`ve tried to express in English.

To tell the truth, eight digit number of placing the year and date side by side which is 20190811 is the prime number.

Thus, when placing the year and the date side by side, and the eight digit number is the prime number, then the year and the date is called the day of the prime number.

As to the number of the prime number in 2019, it`s 19 in all, and the last day of the year in 2019 was...

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【Prime numbers have never been solved】

〈Is today a day of prime numbers?〉

...and the day of the last day of the year in 2019 is also the day of the prime number, but how about the last day of the year in 2020? Is it the day of the prime number?

We have to make sure we divide the number of 20201231 by successive the prime number of 2,3,5,7,11,13, in turn and to be unable to divide the number of 20201231 by any prime number.

If wanting to judge whether or not N is the prime number, we have to make sure N devided by all the prime number until the square root of N.

It`s a translation word...

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【The prime numbers have never been solved】

〈Is today the day of prime number?〉

To tell the truth, as to the last sentence of the last response is a translantion word for word, so I can`t understand why we have to make sure N divied by all the prime number until the square root of N.

The author said it`s very complicated. Even if using an electric calculator, the greater part of people want to give it up, but we don`t have to be worried about it, for the present day is very useful.

There is a site in internet which judges whether or not it`s the prime number until sixteen digit.

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