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

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Source Work

1986: David Wells: Curious and Interesting Numbers:

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


Mistake

The largest known prime in July $1951$, discovered by J.C.P. Miller and D.J. Wheeler 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 A. Ferrier, 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 if and only if $n$ 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:

\(\ds 180 \times \paren {2^{127} - 1} + 1\) \(=\) \(\ds 180 \times 170\,141\,183\,460\,469\,231\,731\,687\,303\,715\,884\,105\,727 + 1\)
\(\ds \) \(=\) \(\ds 30\,625\,413\,022\,884\,461\,711\,703\,714\,668\,859\,139\,030\,861\)
\(\ds \) \(=\) \(\ds 47 \times 457 \times 1\,913\,636\,609 \times 745\,089\,469\,939\,175\,409\,644\,404\,651\)


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 Ferrier's Number

Thirdly, $\paren {2^{143} + 1} / 17$ has been reported incorrectly.

We have that:

\(\ds 2^{143} + 1\) \(=\) \(\ds 11\,150\,372\,599\,265\,311\,570\,767\,859\,136\,324\,180\,752\,990\,209\)
\(\ds \) \(=\) \(\ds 3 \times 683 \times 2003 \times 2731 \times 6\,156\,182\,033 \times 10\,425\,285\,443 \times 15\,500\,487\,753\,323\)

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 Ferrier found.

It was in fact:

\(\ds \dfrac {2^{148} + 1} {17}\) \(=\) \(\ds \dfrac {356\,811\,923\,176\,489\,970\,264\,571\,492\,362\,373\,784\,095\,686\,657} {17}\)
\(\ds \) \(=\) \(\ds 20 \, 988 \, 936 \, 657 \, 440 \, 586 \,486 \, 151 \, 264 \, 256 \, 610 \, 222 \, 593 \, 863 \, 921\)

which has $44$ digits.


This paragraph has been removed from Curious and Interesting Numbers, 2nd ed. of $1997$.


Sources

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