# The Brouwer-Heyting Sequence

Consider the sequence which is ZERO in each position except for a ONE in the first position in which the sequence of all the digits 0123456789 finishes occurring in order in the decimal expansion of pi .

QUESTION: What is the limit of the sequence?

• Classically: ZERO -- eventually the sequence is all zeroes.

The "Principle of the excluded middle".

• Constructively/Intuitionistically: need to determine when/if the "1" occurs.

It seemed impossible 90 years ago that it would ever be "well defined".

• Now: it converges to ZERO.

For N-10>17,387,594,880 the sequence is ZERO.
Did it always converge?

Did it start to converge in July 1997?

Some interesting digit sequences

0123456789 : from 17,387,594,880-th of pi
0123456789 : from 26,852,899,245-th of pi
0123456789 : from 30,243,957,439-th of pi
0123456789 : from 34,549,153,953-th of pi
0123456789 : from 41,952,536,161-th of pi
0123456789 : from 43,289,964,000-th of pi

9876543210 : from 21,981,157,633-th of pi
9876543210 : from 29,832,636,867-th of pi
9876543210 : from 39,232,573,648-th of pi
9876543210 : from 42,140,457,481-th of pi
9876543210 : from 43,065,796,214-th of pi

09876543210 : from 42,321,758,803-th of pi
27182818284 : from 45,111,908,393-th of pi

0123456789 : from 6,214,876,462-th of 1/pi
01234567890 : from 50,494,465,695-th of 1/pi

9876543210 : from 15,603,388,145-th of 1/pi
9876543210 : from 51,507,034,812-th of 1/pi

999999999999 : from 12,479,021,132-th of 1/pi

(First digit '3' for pi or '0' for 1/pi is not included in the above count.)