Concatenation, Squares & Sums [Part 1]

(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

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About benvitalis

math grad - Interest: Number theory
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