**(1)** __Of the form, AB = A || B = A^k + B^k__

**1,233** = 12 || 33 = 12^2 + 33^2

1000*a + 100*b + 33 = (10*a + b)^2 + 33^2

__The only acceptable solutions are__:

a = 1 b = 2

a = 8 b = 8

**8,833** = 88 || 33 = 88^2 + 33^2

Numbers of the form aabb such that aabb = aa || bb = (aa)^2 + (bb)^2

1100 a + 11 b = 121 a^2 + 121 b^2

__Integer solutions__:

a = 0, b = 0

a = 8, b = 3

**10,100** = 10 || 100 = 10^2 + 100^2

**990,100** = 990 || 100 = 990^2 + 100^2

**5,882,353** = 588 || 2353 = 588^2 + 2353^2

**94,122,353** = 9412 || 2353 = 9412^2 + 2353^2

And 9,412 + 588 = 10,000.

**2,584,043,776** = 25840 || 43776 = 25840^2 + 43776^2

**7,416,043,776** = 74160 || 43776 = 74160^2 + 43776^2

and 74160 + 25840 = 100,000.

**116,788,321,168** = 116788 || 321168 = 116788^2 + 321168^2

**(2)** __Of the form, ABC = A || B || C = A^k + B^k + C^k__

**2,213** = 2 || 2 || 13 = 2^3 + 2^3 + 13^3 (__2213 a prime number__)

**221,859** = 22 || 18 || 59 = 22^3 + 18^3 + 59^3

**166,500,333** = 166 || 500 || 333 = 166^3 + 500^3 + 333^3

**(2a) With digits of the number reversed**

**121,184** = 48^3 + 11^3 + 21^3

**444,664** = 46^3 + 64^3 + 44^3

**605,567** = 76^3 + 55^3 + 06^3

**(3)** __Of the form, AB = A || B = A^k – B^k__

**33,346,668** = 3334 || 6668 = 6668^2 – 3334^2

**(4)** __Of the form, AB = A || B = B^k – A^k__

**48** = 4 || 8 = 8^2 – 4^2

**3,468** = 34 || 68 = 68^2 – 34^2

**416,768** = 416 || 768 = 768^2 – 416^2