Taylor - Theoretic Arithmetic.pdf - Platonic Philosophy

These numbers are as follows:
In the first place 2 X 5 and +1=11, 2 )( 23 and + 1=47,
2X 191 and +1=383, 2X 1535 and +1=3071, 2X 12287
and + 1=24575, and 2 X 98303 and +1=196607. And thus
the double of the first number in each rank added to unity is
equal to the second number in the same rank.
In the next place, the first number of the first rank multiplied
by 4 and added to 3, will be equal to the first number of
the second rank. Thus also the second number 11 X 4 and added
to 3=47 the second number of the second rank. But 23 >< 8
and +7=191, 47x8 and +7=383, 191x8 and +7=1535,
383x8 and +7=3071, 1535x8 and +7=12287, 3071x8
and +7=24575, 12287x8 and +7=98303, and 24575~ 8
and +7= 196607.
Again, 47x4 and +3=191, 383x4 and +3=1535, 3071
X 4 and +3=12287, and 24575 >< 4 and +3=98303.
1151 73727-
Again, T=16, with a remainder 15. -64, with a
4718591-
remainder 63. 3 -64, and the remainder is 63. And
301989887-
7 5 -64, with the same remainder 63. And so of the
rest ad infin. the quotient and remainder being always 64
and 63.
Hence infinite series of all these may easily be obtained, viz.
of the two first terms in each rank, the first rank excepted, and
of the third term in each rank the two first ranks excepted. For
23+7+7+7, kc. ad infin.