MathLabs

Problem 6

The function f(x,y)f(x,y) satisfies f(0,y)=y+1,f(0,y) = y+1, f(x+1,0)=f(x,1),f(x+1,0) = f(x,1), f(x+1,y+1)=f(x,f(x+1,y))f(x+1,y+1) = f\big(x, f(x+1,y)\big) for all non-negative integers x,yx,y. Determine f(4,1981)f(4,1981).
Step 7 of 7: State the final value
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f(4,1981)=22⋅⋅2⏟1984 twos−3f(4,1981) = \underbrace{2^{2^{\cdot^{\cdot^{2}}}}}_{1984 \text{ twos}} - 3
Detailed analysis

Subtracting 33 from the tower found in Step 6 gives the requested value f(4,1981)f(4,1981), an astronomically large number expressed exactly as a power tower of 19841984 twos minus 33.