Problem of 1556

In 1556, NiccoloÁ Tartaglia of Brescia claimed that the sums 1 + 2 + 4, 1 + 2 + 4 + 8, 1 + 2 + 4 + 8 + 16, etc. are alternately prime and com- posite. Show that his conjecture is false.

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`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
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Comments

The sum $\displaystyle \sum_{k=0}^n 2^k = 2^{n+1} -1$ .

If we look at the remainders when those powers of 2 are divided by 7, we observe a pattern

$n$	$2^n$	$2^n \mod 7$
2	4	4
3	8	1
4	16	2
5	32	4
6	64	1
$\vdots$	$\vdots$	$\vdots$

We see that this pattern repeats in a cycle of 3.

Tartaglia conjectured that for odd $n$ , $2^n-1$ is a prime.

However, since $2^n$ leaves a remainder of 1 every third time, every sixth time this remainder will occur at an odd number (for $2^{6k+3}, k \in \mathbb{N}$ ) and $2^n-1$ will be divisible by 7.

The first time this occurs is for $1+2+\cdots+128+512$ and infinitely many times after that.

Henry U - 2 years, 4 months ago

Correct

chakravarthy b - 2 years, 3 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$