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

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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


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Sources

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