Left-Truncatable Prime/Examples/357,686,312,646,216,567,629,137

Theorem

The largest left-truncatable prime is $357 \, 686 \, 312 \, 646 \, 216 \, 567 \, 629 \, 137$.

Proof

First it is demonstrated that $357 \, 686 \, 312 \, 646 \, 216 \, 567 \, 629 \, 137$ is indeed a left-truncatable prime:

$\ds 357 \, 686 \, 312 \, 646 \, 216 \, 567 \, 629 \, 137$

$\ds $

is prime

$\ds 57 \, 686 \, 312 \, 646 \, 216 \, 567 \, 629 \, 137$

$\ds $

is prime

$\ds 7 \, 686 \, 312 \, 646 \, 216 \, 567 \, 629 \, 137$

$\ds $

is prime

$\ds 686 \, 312 \, 646 \, 216 \, 567 \, 629 \, 137$

$\ds $

is prime

$\ds 86 \, 312 \, 646 \, 216 \, 567 \, 629 \, 137$

$\ds $

is prime

$\ds 6 \, 312 \, 646 \, 216 \, 567 \, 629 \, 137$

$\ds $

is prime

$\ds 312 \, 646 \, 216 \, 567 \, 629 \, 137$

$\ds $

is prime

$\ds 12 \, 646 \, 216 \, 567 \, 629 \, 137$

$\ds $

is prime

$\ds 2 \, 646 \, 216 \, 567 \, 629 \, 137$

$\ds $

is prime

$\ds 646 \, 216 \, 567 \, 629 \, 137$

$\ds $

is prime

$\ds 46 \, 216 \, 567 \, 629 \, 137$

$\ds $

is prime

$\ds 6 \, 216 \, 567 \, 629 \, 137$

$\ds $

is prime

$\ds 216 \, 567 \, 629 \, 137$

$\ds $

is prime

$\ds 16 \, 567 \, 629 \, 137$

$\ds $

is prime

$\ds 6 \, 567 \, 629 \, 137$

$\ds $

is prime

$\ds 567 \, 629 \, 137$

$\ds $

is prime

$\ds 67 \, 629 \, 137$

$\ds $

is the $3 \, 986 \, 726$th prime

$\ds 7 \, 629 \, 137$

$\ds $

is the $516 \, 434$th prime

$\ds 629 \, 137$

$\ds $

is the $51 \, 275$th prime

$\ds 29 \, 137$

$\ds $

is the $3167$th prime

$\ds 9 \, 137$

$\ds $

is the $1133$rd prime

$\ds 137$

$\ds $

is the $33$rd prime

$\ds 37$

$\ds $

is the $12$th prime

$\ds 7$

$\ds $

is the $4$th prime

This theorem requires a proof.
In particular: It remains to be shown that it is the largest such.
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Sources

1977: I.O. Angell and H.J. Godwin: On Truncatable Primes (Math. Comp. Vol. 31: pp. 265 – 267) www.jstor.org/stable/2005797
1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $357,686,312,646,216,567,629,137$
1987: C. Caldwell: Truncatable primes (J. Recr. Math. Vol. 19, no. 1: pp. 30 – 33)
1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $357,686,312,646,216,567,629,137$