Goldbach's Lesser Conjecture/5993

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Goldbach's Lesser Conjecture: $5993$ is a Stern Number

The number $5993$ cannot be represented in the form:

$5993 = 2 a^2 + p$

where:

$a \in \Z_{\ge 0}$ is a non-negative integer
$p$ is a prime number.


Proof

It will be shown that for all $a$ such that $2 a^2 \le 5993$, it is never the case that $5993 - 2 a^2$ is prime.


Thus:

\(\displaystyle 5993 - 2 \times 0^2\) \(=\) \(\displaystyle 5993\) which is composite: $5993 = 13 \times 461$
\(\displaystyle 5993 - 2 \times 1^2\) \(=\) \(\displaystyle 5991\) which is composite: $5991 = 3 \times 1997$
\(\displaystyle 5993 - 2 \times 2^2\) \(=\) \(\displaystyle 5985\) which is composite: $5985 = 3^2 \times 5 \times 7 \times 19$
\(\displaystyle 5993 - 2 \times 3^2\) \(=\) \(\displaystyle 5975\) which is composite: $5975 = 5^2 \times 239$
\(\displaystyle 5993 - 2 \times 4^2\) \(=\) \(\displaystyle 5961\) which is composite: $5961 = 3 \times 1987$
\(\displaystyle 5993 - 2 \times 5^2\) \(=\) \(\displaystyle 5943\) which is composite: $5943 = 3 \times 7 \times 283$
\(\displaystyle 5993 - 2 \times 6^2\) \(=\) \(\displaystyle 5921\) which is composite: $5921 = 31 \times 191$
\(\displaystyle 5993 - 2 \times 7^2\) \(=\) \(\displaystyle 5895\) which is composite: $5895 = 3^2 \times 5 \times 131$
\(\displaystyle 5993 - 2 \times 8^2\) \(=\) \(\displaystyle 5865\) which is composite: $5865 = 3 \times 5 \times 17 \times 23$
\(\displaystyle 5993 - 2 \times 9^2\) \(=\) \(\displaystyle 5831\) which is composite: $5831 = 7^3 \times 17$
\(\displaystyle 5993 - 2 \times 10^2\) \(=\) \(\displaystyle 5793\) which is composite: $5793 = 3 \times 1931$
\(\displaystyle 5993 - 2 \times 11^2\) \(=\) \(\displaystyle 5751\) which is composite: $5751 = 3^4 \times 71$
\(\displaystyle 5993 - 2 \times 12^2\) \(=\) \(\displaystyle 5705\) which is composite: $5705 = 5 \times 7 \times 163$
\(\displaystyle 5993 - 2 \times 13^2\) \(=\) \(\displaystyle 5655\) which is composite: $5655 = 3 \times 5 \times 13 \times 29$
\(\displaystyle 5993 - 2 \times 14^2\) \(=\) \(\displaystyle 5601\) which is composite: $5601 = 3 \times 1867$
\(\displaystyle 5993 - 2 \times 15^2\) \(=\) \(\displaystyle 5543\) which is composite: $5543 = 23 \times 241$
\(\displaystyle 5993 - 2 \times 16^2\) \(=\) \(\displaystyle 5481\) which is composite: $5481 = 3^3 \times 7 \times 29$
\(\displaystyle 5993 - 2 \times 17^2\) \(=\) \(\displaystyle 5415\) which is composite: $5415 = 3 \times 5 \times 19^2$
\(\displaystyle 5993 - 2 \times 18^2\) \(=\) \(\displaystyle 5345\) which is composite: $5345 = 5 \times 1069$
\(\displaystyle 5993 - 2 \times 19^2\) \(=\) \(\displaystyle 5271\) which is composite: $5271 = 3 \times 7 \times 251$
\(\displaystyle 5993 - 2 \times 20^2\) \(=\) \(\displaystyle 5193\) which is composite: $5193 = 3^2 \times 577$
\(\displaystyle 5993 - 2 \times 21^2\) \(=\) \(\displaystyle 5111\) which is composite: $5111 = 19 \times 269$
\(\displaystyle 5993 - 2 \times 22^2\) \(=\) \(\displaystyle 5025\) which is composite: $5025 = 3 \times 5^2 \times 67$
\(\displaystyle 5993 - 2 \times 23^2\) \(=\) \(\displaystyle 4935\) which is composite: $4935 = 3 \times 5 \times 7 \times 47$
\(\displaystyle 5993 - 2 \times 24^2\) \(=\) \(\displaystyle 4841\) which is composite: $4841 = 47 \times 103$
\(\displaystyle 5993 - 2 \times 25^2\) \(=\) \(\displaystyle 4743\) which is composite: $4743 = 3^2 \times 17 \times 31$
\(\displaystyle 5993 - 2 \times 26^2\) \(=\) \(\displaystyle 4641\) which is composite: $4641 = 3 \times 7 \times 13 \times 17$
\(\displaystyle 5993 - 2 \times 27^2\) \(=\) \(\displaystyle 4535\) which is composite: $4535 = 5 \times 907$
\(\displaystyle 5993 - 2 \times 28^2\) \(=\) \(\displaystyle 4425\) which is composite: $4425 = 3 \times 5^2 \times 59$
\(\displaystyle 5993 - 2 \times 29^2\) \(=\) \(\displaystyle 4311\) which is composite: $4311 = 3^2 \times 479$
\(\displaystyle 5993 - 2 \times 30^2\) \(=\) \(\displaystyle 4193\) which is composite: $4193 = 7 \times 599$
\(\displaystyle 5993 - 2 \times 31^2\) \(=\) \(\displaystyle 4071\) which is composite: $4071 = 3 \times 23 \times 59$
\(\displaystyle 5993 - 2 \times 32^2\) \(=\) \(\displaystyle 3945\) which is composite: $3945 = 3 \times 5 \times 263$
\(\displaystyle 5993 - 2 \times 33^2\) \(=\) \(\displaystyle 3815\) which is composite: $3815 = 5 \times 7 \times 109$
\(\displaystyle 5993 - 2 \times 34^2\) \(=\) \(\displaystyle 3681\) which is composite: $3681 = 3^2 \times 409$
\(\displaystyle 5993 - 2 \times 35^2\) \(=\) \(\displaystyle 3543\) which is composite: $3543 = 3 \times 1181$
\(\displaystyle 5993 - 2 \times 36^2\) \(=\) \(\displaystyle 3401\) which is composite: $3401 = 19 \times 179$
\(\displaystyle 5993 - 2 \times 37^2\) \(=\) \(\displaystyle 3255\) which is composite: $3255 = 3 \times 5 \times 7 \times 31$
\(\displaystyle 5993 - 2 \times 38^2\) \(=\) \(\displaystyle 3105\) which is composite: $3105 = 3^2 \times 5 \times 69$
\(\displaystyle 5993 - 2 \times 39^2\) \(=\) \(\displaystyle 2951\) which is composite: $2951 = 13 \times 227$
\(\displaystyle 5993 - 2 \times 40^2\) \(=\) \(\displaystyle 2793\) which is composite: $2793 = 3 \times 7^2 \times 19$
\(\displaystyle 5993 - 2 \times 41^2\) \(=\) \(\displaystyle 2631\) which is composite: $2631 = 3 \times 877$
\(\displaystyle 5993 - 2 \times 42^2\) \(=\) \(\displaystyle 2465\) which is composite: $2465 = 5 \times 17 \times 29$
\(\displaystyle 5993 - 2 \times 43^2\) \(=\) \(\displaystyle 2295\) which is composite: $2295 = 3^3 \times 5 \times 17$
\(\displaystyle 5993 - 2 \times 44^2\) \(=\) \(\displaystyle 2121\) which is composite: $2121 = 3 \times 7 \times 101$
\(\displaystyle 5993 - 2 \times 45^2\) \(=\) \(\displaystyle 1943\) which is composite: $1943 = 29 \times 67$
\(\displaystyle 5993 - 2 \times 46^2\) \(=\) \(\displaystyle 1761\) which is composite: $1761 = 3 \times 587$
\(\displaystyle 5993 - 2 \times 47^2\) \(=\) \(\displaystyle 1575\) which is composite: $1575 = 3^2 \times 5^2 \times 7$
\(\displaystyle 5993 - 2 \times 48^2\) \(=\) \(\displaystyle 1385\) which is composite: $1385 = 5 \times 277$
\(\displaystyle 5993 - 2 \times 49^2\) \(=\) \(\displaystyle 1191\) which is composite: $1191 = 3 \times 397$
\(\displaystyle 5993 - 2 \times 50^2\) \(=\) \(\displaystyle 993\) which is composite: $993 = 3 \times 331$
\(\displaystyle 5993 - 2 \times 51^2\) \(=\) \(\displaystyle 791\) which is composite: $791 = 7 \times 113$
\(\displaystyle 5993 - 2 \times 52^2\) \(=\) \(\displaystyle 585\) which is composite: $585 = 3^2 \times 5 \times 13$
\(\displaystyle 5993 - 2 \times 53^2\) \(=\) \(\displaystyle 375\) which is composite: $375 = 3 \times 5^3$
\(\displaystyle 5993 - 2 \times 54^2\) \(=\) \(\displaystyle 161\) which is composite: $161 = 7 \times 23$

That exhausts all $a$ such that $2 a^2 \le 5993$, as $2 \times 55^2 = 6050$.

Hence the result.

$\blacksquare$


Source of Name

This entry was named for Christian Goldbach.


Sources