180 x (2^127 - 1) + 1/Mistake

Source Work

 * The Dictionary
 * $180 \times \paren {2^{127} - 1} + 1$
 * $180 \times \paren {2^{127} - 1} + 1$

Mistake

 * The largest known prime in July $1951$, discovered by and  of Cambridge University on the EDSAC. They had a test for numbers of the form $k \times \text M_{127} + 1$ where $\text M_{127}$ is the $127$th Mersenne number. This was the largest prime found.


 * In the same month, using a desk calculator only, showed the primality of $\paren {2^{143} + 1} / 17$.

Correction
There are a number of problems here.

Definition of Mersenne Number
For a start, a number of the form $\text M_n = 2^n - 1$ is usually only defined to be a Mersenne number $n$ is prime.

Hence, as $127$ is the $31$st prime number, $M_{127}$ is generally identified as the $31$st Mersenne number.

Non-Primality of $180 \times \paren {2^{127} - 1} + 1$
Secondly, $180 \times \paren {2^{127} - 1} + 1$ is not actually a prime.

We have:

The number in question is actually:
 * $180 \times \paren {2^{127} - 1}^2 + 1$

which evaluates to:


 * $5 \, 210 \, 644 \, 015 \, 679 \, 228 \, 794 \, 060 \, 694 \, 325 \, 390 \, 955 \, 853 \, 335 \, 898 \, 483 \, 908 \, 056 \, 458 \, 352 \, 183 \, 851 \, 018 \, 372 \, 555 \, 735 \, 221$

which has $79$ digits.

Incorrect reporting of 's Number
Thirdly, $\paren {2^{143} + 1} / 17$ has been reported incorrectly.

We have that:

and so is not actually a multiple of $17$.

But interestingly:
 * $\paren {2^{143} + 1} / 17 = 655\,904\,270\,545\,018\,327\,692\,227\,008\,019\,069\,456\,058\,247$ remainder $10$

and $655\,904\,270\,545\,018\,327\,692\,227\,008\,019\,069\,456\,058\,247$ is indeed prime, and has $42$ digits.

However, a search of the online literature identifies for us what number it was that found.

It was in fact:

which has $44$ digits.

This paragraph has been removed from of $1997$.