Dragan K

If n = 2021*10^2021

and m = n^n

then how can we find the first digit of m?

x + y*z = 3703

y + x*z = 575

z + x*y = 245

How many numbers are there with such properties?

a1^a1 + a2^a2 + ... + an^an = t^2

a1 ... an and t are all different positive integers and n>1.

I am looking for solution with minimal n.

Find length of x and y, exact values:

http://i524.photobucket.com/albums/cc321/Krejakovi...

x and y are on the same line.

Accidentally I find that:

11101^2 = 11100^2 + 149^2

Is there any other case (not congruent with this) of Pythagorean triple in base ten, where two sides can be written only with 0's and 1's?

There are 3 concentric circle with radius 1, 2 and 3, as at the pic:

http://i524.photobucket.com/albums/cc321/Krejakovi...

The same right triangle can fit on the circles as shown at the picture, with hypotenuse as chord of the one circle and right angle vertex on another circle.

Find hypotenuse x?

I accidentally find that

(4 + 3^3)*(4 + 14^3) = 4 + 44^3

Is there any more solutions of

(4 + a^3)*(4 + b^3) = 4 + c^3

where a, b, c are positive integers?

If A^3 + B^3 = C^3 - 2

and A, B, C are positive integers

then A or B must be equal to C-2.

Prove or find counterexample.

If A = (12*n^2+18*n+7)^2

and B = (6n+4)^(6n+4) + (6n+5)^(6n+5)

then A divides B for any natural n.

Number is useful if it can be express as:

s = a1^b1 + a2^b2 + a3^b3 + ... + a(n)^b(n)

where a(i) is consecutive positive integers and b(i) is 2 or 3.

For example, numbers

5^2 + 6^2 + 7^3 + 8^3 = 916, or

4^2 + 5^3 + 6^2 = 177, or

3^3 = 27

are useful numbers.

Number is useless if it can not be express in that way.

Question is - find the largest useless number?

a^3 + b^3 + c^3 + d^3 = 13*a*b*c*d

a,b,c,d are distinct positive integers, can you find them?

a^4 + b^4 ≡ 11 mod (a^2+b^2)

Is there any solution for integer a and b?

Number 449 is a prime number.

Is there any other prime in the form of 449999...999?

(Or if you prefer 45*10^n - 1)

Product of m consecutive positive integers is equal to product of n consecutive positive integers.

Examples:

5*6*7 = 14*15

2*3*4*5 = 4*5*6

19*20*21*22 = 55*56*57

Can you find more examples?

(Must tell that I'm not interested in trivial solutions with m=n, or like 1*2*3*...*n = 2*3*...*n)

a^a + b^b = c^2

a, b, c are positive integers.

Can you make square by concatenating number 25 and some other square?

So, we are looking for solutions of:

concatenate(25; a^2) = b^2

(a and b are positive integers)

Find such example or prove it's impossible?