Graham number with Javascript | A. Matías Quezada

Following this video (in spanish) I learned the arrow notation in math:

3↑3 = 3^3^3 = 3^27 = 7_625_597_484_987

That it gets more brutal the more arrows we add

3↑↑3 = 3↑3↑3 = 3↑7_625_597_484_987
3↑↑↑3 = 3↑↑3↑↑3 = 3↑3↑3↑3↑3↑3↑3↑3... (repeat 7_625_597_484_987 times)

So I wondered how far can a computer take such big calculations, for this I created a function arrow to represent the operation

function arrow(base, exponent) {
    let value = base;

    for (let i = 0; i < exponent; i++) {
        value = value ** base;
    }

    return value;
}

arrow(3, 3) // 7625597484987
arrow(3, 3).toString().length //13

Looks good, the number and calculation for single arrow is correct, let's make the function to handle multiple arrow expressions (3↑↑3 and 3↑↑↑3), for this the amount of arrows will be a new parameter at the end.

function manyArrows(base, exponent, amount = 1) {
    if (amount === 1) {
        // only one arrow, use the previous function
        return arrow(base, exponent);
    }

    let value = base;

    for (let i = 0; i < exponent; i++) {
        value = value ** manyArrows(base, exponent, amount - 1)
    }

    return value;
}

manyArrows(2, 2, 1) // 16
manyArrows(2, 2, 2) // 1.157920892373162e+77
manyArrows(3, 3, 1) // 7625597484987

Good, it works as expected, let's push and get the value for 3↑↑3

manyArrows(3, 3, 2) // Infinity

Hm... we've reached the top limit for Javascript's number type. But wait, there was a proposal for a new feature of BigInt that would add a new numeric type to handle (duh) big integers:

Yep, and it's ready to be used, we just need to add n at the end of a number to make it a BigInt 🎉

manyArrows(3n, 3n, 1) // 7625597484987n
manyArrows(3n, 3n, 2)

// Uncaught RangeError: Maximum BigInt size exceeded
//     at manyArrows (<anonymous>:8:19)
//     at <anonymous>:1:1

Well that's it, we've reached Javascript limits for integer calculations 🤷

If you wondered what this has to do with Graham's number well that is...

let g = [4]

for (let i = 1; i <= 64; i++) {
    let previousLayer = g[i - 1]
    g[i] = manyArrows(3, 3, previousLayer)
}

const GRAHAM_NUMBER = g[64]

let g1 = manyArrows(3, 3, 4)
let g2 = manyArrows(3, 3, g1)
let g3 = manyArrows(3, 3, g2)
let g4 = manyArrows(3, 3, g3)
let g5 = manyArrows(3, 3, g4)
let g6 = manyArrows(3, 3, g5)
let g7 = manyArrows(3, 3, g6)
let g8 = manyArrows(3, 3, g7)
let g9 = manyArrows(3, 3, g8)
let g10 = manyArrows(3, 3, g9)
let g11 = manyArrows(3, 3, g10)
let g12 = manyArrows(3, 3, g11)
let g13 = manyArrows(3, 3, g12)
let g14 = manyArrows(3, 3, g13)
let g15 = manyArrows(3, 3, g14)
let g16 = manyArrows(3, 3, g15)
let g17 = manyArrows(3, 3, g16)
let g18 = manyArrows(3, 3, g17)
let g19 = manyArrows(3, 3, g18)
let g20 = manyArrows(3, 3, g19)
let g21 = manyArrows(3, 3, g20)
let g22 = manyArrows(3, 3, g21)
let g23 = manyArrows(3, 3, g22)
let g24 = manyArrows(3, 3, g23)
let g25 = manyArrows(3, 3, g24)
let g26 = manyArrows(3, 3, g25)
let g27 = manyArrows(3, 3, g26)
let g28 = manyArrows(3, 3, g27)
let g29 = manyArrows(3, 3, g28)
let g30 = manyArrows(3, 3, g29)
let g31 = manyArrows(3, 3, g30)
let g32 = manyArrows(3, 3, g31)
let g33 = manyArrows(3, 3, g32)
let g34 = manyArrows(3, 3, g33)
let g35 = manyArrows(3, 3, g34)
let g36 = manyArrows(3, 3, g35)
let g37 = manyArrows(3, 3, g36)
let g38 = manyArrows(3, 3, g37)
let g39 = manyArrows(3, 3, g38)
let g40 = manyArrows(3, 3, g39)
let g41 = manyArrows(3, 3, g40)
let g42 = manyArrows(3, 3, g41)
let g43 = manyArrows(3, 3, g42)
let g44 = manyArrows(3, 3, g43)
let g45 = manyArrows(3, 3, g44)
let g46 = manyArrows(3, 3, g45)
let g47 = manyArrows(3, 3, g46)
let g48 = manyArrows(3, 3, g47)
let g49 = manyArrows(3, 3, g48)
let g50 = manyArrows(3, 3, g49)
let g51 = manyArrows(3, 3, g50)
let g52 = manyArrows(3, 3, g51)
let g53 = manyArrows(3, 3, g52)
let g54 = manyArrows(3, 3, g53)
let g55 = manyArrows(3, 3, g54)
let g56 = manyArrows(3, 3, g55)
let g57 = manyArrows(3, 3, g56)
let g58 = manyArrows(3, 3, g57)
let g59 = manyArrows(3, 3, g58)
let g60 = manyArrows(3, 3, g59)
let g61 = manyArrows(3, 3, g60)
let g62 = manyArrows(3, 3, g61)
let g63 = manyArrows(3, 3, g62)

const GRAHAM_NUMBER = manyArrows(3, 3, g63)

In cases like this the expression "more than atoms in the universe" falls short.