Are 8 and 9 the only continuous powers?

If an integer is in the form of m n, it is called a complete power, where m and n are integers, and n >: 1.

It is generally assumed that 8 = 2 3, 9 = 3 2 is the only complete power continuous integer.

Unsolved problem 2:

There are infinitely many twin prime numbers, right?

A prime number is an integer greater than 1, and there are no other positive integers except 1 and itself.

Twin prime numbers are two prime numbers with a difference of 2. For example, 17 and 19 are twin prime numbers.

Unresolved problem 3:

Is there a cuboid with integer sides and diagonal lines?

By cuboid, we mean a solid with six rectangular faces. This common figure is also called a cuboid.

Diagonal lines of cuboids include plane diagonal lines and body diagonal lines. Diagonal lines of faces connect opposite vertices of faces. The volume diagonal (or space diagonal) connects the opposite vertices of the cuboid.

Unsolved problem 4:

Can a closed plane curve have more than one isosceles point?

A line segment connecting two points on a curve is called a chord.

A point in a closed convex plane curve is called an isosceles point, which means that all chords passing through this point have the same length. For example, the center of the circle is the isosceles point of the circle.

I don't know yet. Are there two closed curves with different isosceles points?

Unsolved problem 5:

Is every even number greater than 2 the sum of two prime numbers?

A prime number is an integer greater than 1, and it has only 1 and itself as a positive divisor.

For example, even number 50 is the sum of two prime numbers 3 and 47.

Unresolved problem 6:

Are there infinite Fibonacci prime numbers?

A prime number is an integer greater than 1, and it has only 1 and itself as a positive divisor.

Fibonacci numbers are numbers in the following sequence:

1,1,2, 3, 5, 8,13, 21,34, 55, 89,144, ... each item is the sum of the first two items.

Unsolved problem 7:

Is there a Rubik's Cube knight traveling on the 8 X 8 chessboard?

A knight's journey on the chessboard is a knight's moving sequence, so that each square on the chessboard can be visited exactly once.

Let the continuous squares be numbered from 1 to 64. If the square generated by the trip is a Rubik's cube, the trip is called Rubik's cube trip.

Rubik's cube is a square arrangement of numbers, so that the sum of the numbers on each row, column and two main diagonals is the same number (Rubik's cube constant).

Half Rubik's Cube Knight Travel Known. The sum of numbers in each row and column is the same number, but the sum of diagonal lines is not that number.

Unresolved problems 8

Is π+e an irrational number?

The π number is the ratio of the circumference to the diameter of a circle.

The number e is the base of natural logarithm, which is approximately equal to 2.7 1828. It is a unique number-the derivative of x is X.

A rational number is a number that can be expressed as the ratio of two integers. All other real numbers are called irrational numbers.

E irrational number and π irrational number are known, but I don't know whether their sum is irrational.

Unresolved problem 9:

Let k>0, all rectangles with sides of 1/ k and1(k+1) can fill the unit square of 1X 1?

Unresolved problems 10:

Let n be an integer greater than 1. Is it necessary to satisfy the integer x, y and z of 4/n =1/x+1/y+1/z?

Let x be an integer, and the score in the form of 1/ x is called Egyptian score.

We want to know whether 4/n is always the sum of three Egyptian scores, that is, n >: 1.

Unresolved problems 1 1

Do you have odd perfect numbers? )

A perfect number is a positive integer and the sum of all its positive integer factors (except itself) is equal to itself.

For example, 28 is complete because 28= 1+2+4+7+ 14.

Unresolved problems 12

Is every tree beautiful?

A graph is a set of points (called vertices) and a set of lines (called edges) connecting these vertices. )

A tree is a graph with the following characteristics: from any vertex to another vertex, there is a unique path along the edge.

A graph is called a graceful graph. If you can mark n vertices with integers from 1 to n, and then mark each edge with the difference between numbers, then each edge will get a different label.

For example, a beautiful label for a tree with nine vertices:

(5)( 1)-(4)& amp； My q# F4 r5

/ /

(7) - (3) - (9) - (2)8 g- a9 y6 z% J/ t，？ ; T

\ \9 k% P4 K8 Y. f3 M

(6)(8)5z5l & amp； z* P4 {+ h0 Y+ ]。 e

The edge label is a number from 1 to 8.

Unresolved problems 13:

Is there a point on the plane that makes the distance between the change point and the four vertices of the unit square reasonable?

A rational number is a number that can be expressed as the ratio of two integers.

A unit square is a square with a side length of 1.

Unresolved problems 14:

What is the value of1/1+1/8+1/27+1/64+1/25+?

The nth term is the reciprocal of n 3.

If the power of 3 is replaced by 2, the sequence sum is (π 2)/6.

If the power of 3 is replaced by 4, the sequence sum is (π 4)/90.

Unresolved problems 15

Is every Mason number a non-square number?

Mason number is a number in the form of 2 p- 1, where p is a prime number.

A prime number is an integer greater than 1, and it has only 1 and itself as a positive divisor.

An integer is called a non-square number, which means that it does not contain the complete square number n 2 (n >; 1) as its factor. .

Unresolved problems 16:

Does each obtuse triangle contain the periodic orbit of billiards trajectory?

Let's assume that billiards bounce back after hitting each face with a reflection angle equal to the incident angle. If it touches a vertex, it will bounce back along the bisector of the vertex. A track is periodic if it returns to its starting point after a limited number of reflections.

Unresolved problems 17:

Is there such a set s in the plane? The set that satisfies each contract in S contains exactly one cell?

Grid points are points with integer coordinates.

Unresolved problems 18:

There are different positive integers a, b, c, d, which satisfy A 5+B 5 = C 5+D 5?

It is known that13+123 = 9.3+103 and133.4+134.4 = 59.4+158.4, but they are all quintic.

Other typical results

27^5+84^5+ 1 10^5+ 133^5= 144^5 2j M9 j v * m % t # h

2682440^4+ 15365639^ 4+ 18796760^4=206 15673^4

9 a) Z( I，p. n: i

Unsolved problem19:: v $ r'}; K2 m+ f，c( z" ~

When disks of the same size are squeezed closer together, will their total area increase? / V5 k7 [。 l. m9 J: J0 g

A disk, we mean a circle and its interior. It is known that the results of two disks are correct. The disks are squeezed together, which means that the distance between all disks becomes smaller after squeezing. The union of a set of disks is the area covered by all disks. Allow disks to overlap. # b7 q！ o，i8 F2 t 1 b- n

1 J！ z2 M9 Q m$ k6 S

Unsolved problem 20: .a7 \ 6s.f: k&; \6 i

Is there an infinite number of prime numbers in the form of n 2+1 7n % c $ m7m & amp； W# V- B% m

5 a，w2 s0 t！ t' m. `: \

Unsolved problem 2 1:

Is every integer greater than 454 the sum of seven or fewer positive cubes?

Unsolved problem 22:" h# f) U0 j) r: g L/ g

Is there a triangle whose sides, midline and area are integers?

The midline of a triangle is the line connecting the vertex with the midpoint of its opposite side.

% T/ q7 s，v% v

Unresolved problem 23:

How to arrange 13 cities on a spherical planet so that the minimum distance between any two cities is as large as possible? + ^.F5 d. n4 L，s

; ? 9 _8 _- y 1 X$ p

Unsolved problem 24:1p; l" Q# J8 S8 D

Is there always a prime number between any two consecutive squares? / Q$ S，b' q2 p( ]6 r

Unresolved problem 25:

Starting from any positive integer. If it is an even number, it is divided equally; If it is odd, triple 1. Repeat this process repeatedly, will you finally get 1?

For example, starting from the 6th, we get: 6,3, 1 0,5,16,8,4,2,1. (S！ q(y0 \ f & amp； A & amp_0 W

7 J8 A) @# _8？ 7 K) _( e

Unsolved problem 26:+ a4 }) {/ R- Q9 |

Given a simple plane closed curve, can you always find four points on this curve as four vertices of a square? i6 L6 _； y； C: r$ x

Unsolved problem 27: 5G9B7a; w e# W {，y

There are integers n and x (where n >；; 7) satisfy n! = x ^ 2- 1？ 0 \/F g % k & amp； i4 j

n！ This means that the integer is multiplied by n from 1. n2 m: s: W3 L，Z

Known 4! + 1=25=5^2, 5! +1=121=12, and 7! + 1=504 1=7 1^2.

Unresolved problem 28:

3 can be written as 1 3+ 1 3+ 1 3 and 4 3+4 3+(-5) 3. Expression 3 is the sum of three (positive or negative) cubes. Is there any other way? # q ' C； k+ K* l9 M

7 \! q； M3 f

Unresolved problem 29:

The sides of triangle A are a 1, a2, a3, and the corresponding sides of triangle B are b 1, b2, b3. What are the necessary and sufficient conditions for the variables a 1, a2, a3 and b 1, b2, b3 to make triangle A fit triangle B?

! I5 V (length: 1 d

Unresolved problem 30:

Is every integer the sum of four cubes?

Here we allow the cubic number to be positive, negative or zero.

For example, 84 = 0 3+41639613+(-41531726) 3+(-82411). ! j8 R2 M$ j6 p ~

For example, I don't know whether 148 is the sum of four cubes. ; \: F( g- o# f

$ c9 @ `+ E3 r" z！ g

Unsolved problem 3 1:

N points can always be found on the plane (there is no 3-point * * * line; Don't you have a 4-point circle? Satisfy every k (0

For example, four points determine six distances. We want one distance to appear only once, another distance to appear twice and the third distance to appear three times. - A9 W$ F5 ]4 v- e，|

So far, n=2, 3, 4, 8 ... has been found. +V6 ~ 0[7m & amp； x6 U 1 k+ b

- E l0 l6 u$ @3 k9 P

Unresolved problem 32:

Can you find three integers x, y, z and satisfy (x+y+z) 3 = XYZ? * y 1 T: }7 G' R6 _- X/ r

(H5 a(U； l8 |)C； M7 d) m7 X* J

Unsolved problem 33:3 N v* j( {! L) n% u

Taking the constant a, there must be a point set with an area of a on the plane, so that it contains the vertices of a triangle with an area of 1

)e t p # _ 1 _ 6 ^( d

Unresolved problem 34:

Is there a finite number of perfect squares consisting of only two different nonzero decimal numbers? . ` 0 v8 }，o" R9 Z' T7 j，@ 1 @。 {

For example, 38 2 =1444, 88 2 = 7744,109 2 =1881,173 2 = 29929. N9 F9 z* q" b

Unresolved problem 35:

N points on a plane are not * * * lines. Is there necessarily a point that is at least on n/3 straight lines determined by these points?

Unresolved problem 36:&; T3 b $ R7 E7 H； n9 V$ [2 Z c

Are there any other values besides 1, 2, 4, n that satisfy that N+ 1 is a prime number?