Titanic Sophie Germain Prime

Theorem

The integer defined as:

$39 \, 051 \times 2^{6001} - 1$

is a titanic prime which is also a Sophie Germain prime:

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$\ds 11820 \, 50794 \, 19125 \, 52383 \, 74423 \, 53078 \, 56017 \, 05024 \, 84819 \, 01689$

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$\ds 74975 \, 95139 \, 68621 \, 89553 \, 48654 \, 81137 \, 72841 \, 27658 \, 52217 \, 40999$

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$\ds 04778 \, 71896 \, 78015 \, 63535 \, 94741 \, 82340 \, 68638 \, 88011 \, 18130 \, 14219$

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$\ds 81435 \, 50235 \, 73607 \, 51980 \, 74200 \, 04306 \, 58030 \, 53360 \, 79821 \, 16678$

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$\ds 32541 \, 21729 \, 72493 \, 53731 \, 27605 \, 59447 \, 95967 \, 46064 \, 11137 \, 07858$

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$\ds 37078 \, 27755 \, 33462 \, 32179 \, 66482 \, 80947 \, 33386 \, 65681 \, 87582 \, 11189$

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$\ds 25630 \, 83169 \, 50526 \, 70023 \, 66301 \, 83449 \, 99960 \, 25913 \, 90035 \, 61496$

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$\ds 03726 \, 62661 \, 50693 \, 56343 \, 90085 \, 30468 \, 46645 \, 69888 \, 03202 \, 50070$

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$\ds 38139 \, 19172 \, 69637 \, 71838 \, 13812 \, 48256 \, 38384 \, 37787 \, 83423 \, 06357$

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$\ds 09062 \, 96393 \, 13908 \, 65400 \, 30048 \, 07291 \, 64958 \, 29772 \, 97828 \, 35273$

$\ds $

$\ds 02603 \, 73947 \, 05739 \, 46904 \, 93564 \, 50661 \, 00172 \, 36892 \, 20285 \, 60354$

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$\ds 58830 \, 25332 \, 20848 \, 80128 \, 32451 \, 94645 \, 21648 \, 78503 \, 66425 \, 73281$

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$\ds 55405 \, 94426 \, 29476 \, 00573 \, 05011 \, 86259 \, 25148 \, 08537 \, 31389 \, 24832$

$\ds $

$\ds 90593 \, 45279 \, 70389 \, 89332 \, 87614 \, 90279 \, 77417 \, 70009 \, 37843 \, 56718$

$\ds $

$\ds 78965 \, 55090 \, 40413 \, 05491 \, 45610 \, 39734 \, 55313 \, 36378 \, 82326 \, 51747$

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$\ds 26323 \, 96872 \, 58800 \, 36097 \, 85595 \, 50576 \, 58179 \, 78961 \, 56439 \, 38001$

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$\ds 61356 \, 42993 \, 82918 \, 89157 \, 64818 \, 24068 \, 61810 \, 98754 \, 13407 \, 25598$

$\ds $

$\ds 81076 \, 88939 \, 65566 \, 79970 \, 94454 \, 12508 \, 20606 \, 03037 \, 82723 \, 11003$

$\ds $

$\ds 86445 \, 85147 \, 95431 \, 68421 \, 48123 \, 63910 \, 96321 \, 63833 \, 76594 \, 77873$

$\ds $

$\ds 36044 \, 25100 \, 46756 \, 76942 \, 21197 \, 98655 \, 69863 \, 08993 \, 13991 \, 54810$

$\ds $

$\ds 29955 \, 71299 \, 30916 \, 19908 \, 66968 \, 53268 \, 78801 \, 17165 \, 95377 \, 09390$

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$\ds 12417 \, 99779 \, 38952 \, 06419 \, 62790 \, 94932 \, 21996 \, 15477 \, 09894 \, 18755$

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$\ds 79741 \, 05192 \, 62661 \, 21081 \, 92384 \, 45257 \, 78675 \, 87928 \, 74768 \, 12218$

$\ds $

$\ds 63148 \, 68786 \, 76854 \, 53862 \, 69957 \, 63612 \, 71978 \, 31119 \, 74476 \, 86496$

$\ds $

$\ds 45065 \, 87748 \, 91053 \, 15072 \, 63384 \, 65410 \, 90174 \, 27502 \, 19115 \, 20006$

$\ds $

$\ds 99485 \, 86281 \, 23536 \, 18641 \, 48374 \, 90557 \, 49920 \, 15285 \, 92211 \, 19416$

$\ds $

$\ds 75209 \, 57766 \, 75409 \, 22211 \, 29543 \, 79999 \, 81129 \, 89523 \, 59262 \, 62800$

$\ds $

$\ds 46942 \, 15484 \, 08243 \, 63610 \, 64351 \, 53563 \, 01617 \, 42451 \, 12051 \, 59183$

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$\ds 34354 \, 13049 \, 42449 \, 46301 \, 59875 \, 51181 \, 09280 \, 53716 \, 57952 \, 29658$

$\ds $

$\ds 01206 \, 92006 \, 20396 \, 63689 \, 45859 \, 75910 \, 58626 \, 38955 \, 88424 \, 79023$

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$\ds 70325 \, 29477 \, 90965 \, 29020 \, 39505 \, 24422 \, 75678 \, 32327 \, 27410 \, 18290$

$\ds $

$\ds 15226 \, 89958 \, 01677 \, 48481 \, 42430 \, 49977 \, 81717 \, 47239 \, 67104 \, 08734$

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$\ds 21063 \, 13953 \, 69197 \, 18416 \, 66197 \, 78782 \, 49199 \, 73757 \, 81152 \, 15777$

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$\ds 88246 \, 98396 \, 88365 \, 29090 \, 59197 \, 96301 \, 79613 \, 87838 \, 71578 \, 75079$

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$\ds 17192 \, 38121 \, 06694 \, 45136 \, 51899 \, 17332 \, 26537 \, 65466 \, 92624 \, 57805$

$\ds $

$\ds 18650 \, 91862 \, 60159 \, 38818 \, 25424 \, 40894 \, 26520 \, 87364 \, 29048 \, 52293$

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$\ds 88924 \, 40043 \, 51$

Proof

At $1812$ digits, it is clear by definition that this prime is titanic.

It was checked that it is a prime number using the "Alpertron" Integer factorisation calculator on $6$th March $2022$.

This took approximately $4.7$ seconds.

To show that it is in fact a Sophie Germain prime, we also need to check that:

$2 \paren {39 \, 051 \times 2^{6001} - 1} + 1 = 39 \, 051 \times 2^{6002} - 1$

is also prime.

It was checked that it is a prime number using the "Alpertron" Integer factorisation calculator on $6$th March $2022$.

This took approximately $4.5$ seconds.

Historical Note

When David Wells documented this entry in his Curious and Interesting Numbers, 2nd ed. of $1997$, this titanic prime was the largest Sophie Germain prime known.

There are now plenty of larger ones known.

Sources

1989: Paulo Ribenboim: The Book of Prime Number Records (2nd ed.)

1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $39,051 \times 2^{6001} - 1$