Sequence of Palindromic Sophie Germain Primes

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Theorem

The number $N = 191 \, 918 \, 080 \, 818 \, 091 \, 909 \, 090 \, 909 \, 190 \, 818 \, 080 \, 819 \, 191$ has the property that:

$N$ is a palindromic Sophie Germain prime
$2 N + 1$ is also a palindromic Sophie Germain prime
$2 \left({2 N + 1}\right) + 1$ is also a palindromic prime, but not a Sophie Germain prime.


Proof

By direct calculation:

\(\ds \) \(\) \(\ds 191 \, 918 \, 080 \, 818 \, 091 \, 909 \, 090 \, 909 \, 190 \, 818 \, 080 \, 819 \, 191\) is palindromic and prime
\(\ds \) \(\) \(\ds 2 \times 191 \, 918 \, 080 \, 818 \, 091 \, 909 \, 090 \, 909 \, 190 \, 818 \, 080 \, 819 \, 191 + 1\)
\(\ds \) \(=\) \(\ds 383 \, 836 \, 161 \, 636 \, 183 \, 818 \, 181 \, 818 \, 381 \, 636 \, 161 \, 638 \, 383\) is palindromic and prime
\(\ds \) \(\) \(\ds 2 \times 383 \, 836 \, 161 \, 636 \, 183 \, 818 \, 181 \, 818 \, 381 \, 636 \, 161 \, 638 \, 383 + 1\)
\(\ds \) \(=\) \(\ds 767 \, 672 \, 323 \, 272 \, 367 \, 636 \, 363 \, 636 \, 763 \, 272 \, 323 \, 276 \, 767\) is palindromic and prime
\(\ds \) \(\) \(\ds 2 \times 767 \, 672 \, 323 \, 272 \, 367 \, 636 \, 363 \, 636 \, 763 \, 272 \, 323 \, 276 \, 767 + 1\)
\(\ds \) \(=\) \(\ds 1 \, 535 \, 344 \, 646 \, 544 \, 735 \, 272 \, 727 \, 273 \, 526 \, 544 \, 646 \, 553 \, 535\) is neither prime nor palindromic

$\blacksquare$


Sources

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