Number Theory Problem No.5

Number Theory Level 4

I am thinking an interesting number α \alpha that satisfy:

1 + b b + a a b + c b + a b a + ( a c ) b = 1 + ( a a b ) b = α 1+b^{b}+a^{ab}+c^{b}+a^{b^{a}}+(ac)^{b}=1+(a^{a}b)^{b}=\alpha .

Find the smallest value of α \alpha if all given numbers are distinct positive integers.


The answer is 1729.

Anh Khoa Nguyễn Ngọc by Anh Khoa Nguyễn Ngọc , 18, Vietnam

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1 solution

Wolfram Mathematica exhaustive search: Flatten [ ParallelTable [ If [ a b a + a a b + ( a c ) b + b b + c b + 1 = ( a a b ) b + 1 , { a , b , c } , Nothing ] , { a , 9 } , { b , 9 } , { c , 9 } ] , 2 ] → ( 2 3 5 ) \text{Flatten}\left[\text{ParallelTable}\left[\text{If}\left[a^{b^a}+a^{a b}+(a c)^b+b^b+c^b+1=\left(a^a b\right)^b+1,\{a,b,c\},\text{Nothing}\right],\{a,9\},\{b,9\},\{c,9\}\right],2\right]\to \left( \begin{array}{ccc} 2 & 3 & 5 \\ \end{array} \right) . Since the digits are distinct, that condition also has been meet. Any other values using larger integer ought to result in even larger answers as there are no subtractions in the expressions.

Wolfram Mathematica exhaustive search means write a little program that saves the triple of values of a,b,and c such that the logical expression (the comparison of the left and right sides of the given expression is itself a logical) when true saves the triple. I suspected that the variables were all less then or equal to 9 and would have tried larger values if the initial search were not successful.

See: Hardy-Ramanujan number.

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