Conway’s chained arrow notation


Conway’s chained arrow notation is a way of writing numbers even larger than those provided by the up arrow notation. We define mnp=m(p+2)n=mpn and mn=mn1=mn. Longer chains are evaluated by

mnp1=mnp
mn1q=mn

and

mnp+1q+1=mn(mnpq+1)q

For example:

332=
3(322)1=
3(322)=
3(3(312)1)=
3(331)=
333=
327=7625597484987

A much larger example is:

3244=
32(3234)3=
32(32(3224)3)3=
32(32(32(3214)3)3)3=
32(32(32(32)3)3)3=
32(32(3293)3)3

Clearly this is going to be a very large number. Note that, as large as it is, it is proceeding towards an eventual final evaluation, as evidenced by the fact that the final number in the chain is getting smaller.

Title Conway’s chained arrow notation
Canonical name ConwaysChainedArrowNotation
Date of creation 2013-03-22 12:58:46
Last modified on 2013-03-22 12:58:46
Owner Henry (455)
Last modified by Henry (455)
Numerical id 8
Author Henry (455)
Entry type Definition
Classification msc 00A05
Synonym chained arrow notation
Synonym chained arrow
Synonym chained-arrow
Synonym chained-arrow notation
Synonym Conway notation
Related topic KnuthsUpArrowNotation