the need for 64 bits

[Mensanator]

I routinely use large numbers in my Collatz Conjecture work.

Really large. As in a quarter million bits. You wouldn't think
that the processor would make all that much difference. But
using the number is a doddle. The real trick is getting there.

There is a limitation that few encounter. In an exponential, the
exponent is limited to 32 bits, else you get an "outrageous
exponent" error. But I assume that was due to integer size,
that if you have 64 bit integers, the limit of the exponent
ought to also be 64 bits.

With my new MacBook replacing my old Windows laptop,
I can try this.

Using this function (which grows really large, really quick)...

def Type12MH(k,i):
    """Find ith, kth Generation Type [1,2] Mersenne Hailstone
       using the closed form equation

    Type12MH(k,i)
    k: generation
    i: member of generation
    returns Hailstone (a)
    """
    ONE = gmpy.mpz(1)
    TWO = gmpy.mpz(2)
    SIX = gmpy.mpz(6)
    NIN = gmpy.mpz(9)
    if (k<1) or (i<1): return 0
    i = gmpy.mpz(i)
    k = gmpy.mpz(k)
    # a = (i-1)*9**(k-1) + (9**(k-1) - 1)//2 + 1
    # return 2**(6*a - 1) - 1
    a = (i-ONE)*NIN**(k-ONE) + (NIN**(k-ONE) - ONE)//TWO + ONE
    return TWO**(SIX*a - ONE) - ONE


i:  1   bits:              5   decimals:            2
i:  2   bits:             29   decimals:            9
i:  3   bits:            245   decimals:           74
i:  4   bits:          2,189   decimals:          659
i:  5   bits:         19,685   decimals:        5,926
i:  6   bits:        177,149   decimals:       53,328
i:  7   bits:      1,594,325   decimals:      479,940
i:  8   bits:     14,348,909   decimals:    4,319,453
i:  9   bits:    129,140,165   decimals:   38,875,064
i: 10   bits:  1,162,261,469   decimals:  349,875,565

… my Windows/32-bit machine can't get past generation 10 without
getting "outrageous exponent".

But with a 64-bit processor, that limitation no longer stops me.

i: 11   bits: 10,460,353,205   decimals:  3,148,880,080
i: 12   bits: 94,143,178,829   decimals: 28,339,920,715

Wow! 94 billion bits! 28 billion decimal digits!

Of course, once one wall falls, you get to go up against the next
one.
For generation 13, I get:

gmp: overflow in mpz type
Abort trap

Hmm, not sure what "overflow" means in this context, but I suspect
it ran out of memory, I probably should have gotten the MacBook Pro
with 8 GB of ram. But then, maybe it wouldn't help.

Regardless, my window of research has gotten slightly larger.