This Demonstration was inspired by a passage in Sylvia Nasar's book, A Beautiful Mind, The Life of Mathematical Genius and Nobel Laureate John Nash, (Touchstone Edition, 2001). Page 80 of that book relates an anecdote about John von Neumann's ability to divine by inspection the smallest integral power of 2 in which a specified integer was at a given position, counting from either left or right. Let the von Neumann exponent be the smallest integer so that contains the target digit in the specified position. This exercise performs that computation, reporting the exponent , the power , and the target integer highlighted in yellow.
To work with variations on the theme of target integer position in integral power expressions, change the base in the program from 2 to other integers greater than zero and construct tables that evaluate the module over various domains. The program used to develop this Demonstration has vast potential for research applications in A New Kind of Science.
The position begins at 2 in order to eliminate the potential of attempting to find zero as a leading integer in a power of 2. You can set the position iterator in this Demonstration arbitrarily high, but that might require additional formatting work for large values.
The term von Neumann exponent was minted for this Demonstration's narrative and is not intended to represent a formal mathematical concept.