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:
\(\ds 5993 - 2 \times 0^2\) | \(=\) | \(\ds 5993\) | which is composite: $5993 = 13 \times 461$ | |||||||||||
\(\ds 5993 - 2 \times 1^2\) | \(=\) | \(\ds 5991\) | which is composite: $5991 = 3 \times 1997$ | |||||||||||
\(\ds 5993 - 2 \times 2^2\) | \(=\) | \(\ds 5985\) | which is composite: $5985 = 3^2 \times 5 \times 7 \times 19$ | |||||||||||
\(\ds 5993 - 2 \times 3^2\) | \(=\) | \(\ds 5975\) | which is composite: $5975 = 5^2 \times 239$ | |||||||||||
\(\ds 5993 - 2 \times 4^2\) | \(=\) | \(\ds 5961\) | which is composite: $5961 = 3 \times 1987$ | |||||||||||
\(\ds 5993 - 2 \times 5^2\) | \(=\) | \(\ds 5943\) | which is composite: $5943 = 3 \times 7 \times 283$ | |||||||||||
\(\ds 5993 - 2 \times 6^2\) | \(=\) | \(\ds 5921\) | which is composite: $5921 = 31 \times 191$ | |||||||||||
\(\ds 5993 - 2 \times 7^2\) | \(=\) | \(\ds 5895\) | which is composite: $5895 = 3^2 \times 5 \times 131$ | |||||||||||
\(\ds 5993 - 2 \times 8^2\) | \(=\) | \(\ds 5865\) | which is composite: $5865 = 3 \times 5 \times 17 \times 23$ | |||||||||||
\(\ds 5993 - 2 \times 9^2\) | \(=\) | \(\ds 5831\) | which is composite: $5831 = 7^3 \times 17$ | |||||||||||
\(\ds 5993 - 2 \times 10^2\) | \(=\) | \(\ds 5793\) | which is composite: $5793 = 3 \times 1931$ | |||||||||||
\(\ds 5993 - 2 \times 11^2\) | \(=\) | \(\ds 5751\) | which is composite: $5751 = 3^4 \times 71$ | |||||||||||
\(\ds 5993 - 2 \times 12^2\) | \(=\) | \(\ds 5705\) | which is composite: $5705 = 5 \times 7 \times 163$ | |||||||||||
\(\ds 5993 - 2 \times 13^2\) | \(=\) | \(\ds 5655\) | which is composite: $5655 = 3 \times 5 \times 13 \times 29$ | |||||||||||
\(\ds 5993 - 2 \times 14^2\) | \(=\) | \(\ds 5601\) | which is composite: $5601 = 3 \times 1867$ | |||||||||||
\(\ds 5993 - 2 \times 15^2\) | \(=\) | \(\ds 5543\) | which is composite: $5543 = 23 \times 241$ | |||||||||||
\(\ds 5993 - 2 \times 16^2\) | \(=\) | \(\ds 5481\) | which is composite: $5481 = 3^3 \times 7 \times 29$ | |||||||||||
\(\ds 5993 - 2 \times 17^2\) | \(=\) | \(\ds 5415\) | which is composite: $5415 = 3 \times 5 \times 19^2$ | |||||||||||
\(\ds 5993 - 2 \times 18^2\) | \(=\) | \(\ds 5345\) | which is composite: $5345 = 5 \times 1069$ | |||||||||||
\(\ds 5993 - 2 \times 19^2\) | \(=\) | \(\ds 5271\) | which is composite: $5271 = 3 \times 7 \times 251$ | |||||||||||
\(\ds 5993 - 2 \times 20^2\) | \(=\) | \(\ds 5193\) | which is composite: $5193 = 3^2 \times 577$ | |||||||||||
\(\ds 5993 - 2 \times 21^2\) | \(=\) | \(\ds 5111\) | which is composite: $5111 = 19 \times 269$ | |||||||||||
\(\ds 5993 - 2 \times 22^2\) | \(=\) | \(\ds 5025\) | which is composite: $5025 = 3 \times 5^2 \times 67$ | |||||||||||
\(\ds 5993 - 2 \times 23^2\) | \(=\) | \(\ds 4935\) | which is composite: $4935 = 3 \times 5 \times 7 \times 47$ | |||||||||||
\(\ds 5993 - 2 \times 24^2\) | \(=\) | \(\ds 4841\) | which is composite: $4841 = 47 \times 103$ | |||||||||||
\(\ds 5993 - 2 \times 25^2\) | \(=\) | \(\ds 4743\) | which is composite: $4743 = 3^2 \times 17 \times 31$ | |||||||||||
\(\ds 5993 - 2 \times 26^2\) | \(=\) | \(\ds 4641\) | which is composite: $4641 = 3 \times 7 \times 13 \times 17$ | |||||||||||
\(\ds 5993 - 2 \times 27^2\) | \(=\) | \(\ds 4535\) | which is composite: $4535 = 5 \times 907$ | |||||||||||
\(\ds 5993 - 2 \times 28^2\) | \(=\) | \(\ds 4425\) | which is composite: $4425 = 3 \times 5^2 \times 59$ | |||||||||||
\(\ds 5993 - 2 \times 29^2\) | \(=\) | \(\ds 4311\) | which is composite: $4311 = 3^2 \times 479$ | |||||||||||
\(\ds 5993 - 2 \times 30^2\) | \(=\) | \(\ds 4193\) | which is composite: $4193 = 7 \times 599$ | |||||||||||
\(\ds 5993 - 2 \times 31^2\) | \(=\) | \(\ds 4071\) | which is composite: $4071 = 3 \times 23 \times 59$ | |||||||||||
\(\ds 5993 - 2 \times 32^2\) | \(=\) | \(\ds 3945\) | which is composite: $3945 = 3 \times 5 \times 263$ | |||||||||||
\(\ds 5993 - 2 \times 33^2\) | \(=\) | \(\ds 3815\) | which is composite: $3815 = 5 \times 7 \times 109$ | |||||||||||
\(\ds 5993 - 2 \times 34^2\) | \(=\) | \(\ds 3681\) | which is composite: $3681 = 3^2 \times 409$ | |||||||||||
\(\ds 5993 - 2 \times 35^2\) | \(=\) | \(\ds 3543\) | which is composite: $3543 = 3 \times 1181$ | |||||||||||
\(\ds 5993 - 2 \times 36^2\) | \(=\) | \(\ds 3401\) | which is composite: $3401 = 19 \times 179$ | |||||||||||
\(\ds 5993 - 2 \times 37^2\) | \(=\) | \(\ds 3255\) | which is composite: $3255 = 3 \times 5 \times 7 \times 31$ | |||||||||||
\(\ds 5993 - 2 \times 38^2\) | \(=\) | \(\ds 3105\) | which is composite: $3105 = 3^2 \times 5 \times 69$ | |||||||||||
\(\ds 5993 - 2 \times 39^2\) | \(=\) | \(\ds 2951\) | which is composite: $2951 = 13 \times 227$ | |||||||||||
\(\ds 5993 - 2 \times 40^2\) | \(=\) | \(\ds 2793\) | which is composite: $2793 = 3 \times 7^2 \times 19$ | |||||||||||
\(\ds 5993 - 2 \times 41^2\) | \(=\) | \(\ds 2631\) | which is composite: $2631 = 3 \times 877$ | |||||||||||
\(\ds 5993 - 2 \times 42^2\) | \(=\) | \(\ds 2465\) | which is composite: $2465 = 5 \times 17 \times 29$ | |||||||||||
\(\ds 5993 - 2 \times 43^2\) | \(=\) | \(\ds 2295\) | which is composite: $2295 = 3^3 \times 5 \times 17$ | |||||||||||
\(\ds 5993 - 2 \times 44^2\) | \(=\) | \(\ds 2121\) | which is composite: $2121 = 3 \times 7 \times 101$ | |||||||||||
\(\ds 5993 - 2 \times 45^2\) | \(=\) | \(\ds 1943\) | which is composite: $1943 = 29 \times 67$ | |||||||||||
\(\ds 5993 - 2 \times 46^2\) | \(=\) | \(\ds 1761\) | which is composite: $1761 = 3 \times 587$ | |||||||||||
\(\ds 5993 - 2 \times 47^2\) | \(=\) | \(\ds 1575\) | which is composite: $1575 = 3^2 \times 5^2 \times 7$ | |||||||||||
\(\ds 5993 - 2 \times 48^2\) | \(=\) | \(\ds 1385\) | which is composite: $1385 = 5 \times 277$ | |||||||||||
\(\ds 5993 - 2 \times 49^2\) | \(=\) | \(\ds 1191\) | which is composite: $1191 = 3 \times 397$ | |||||||||||
\(\ds 5993 - 2 \times 50^2\) | \(=\) | \(\ds 993\) | which is composite: $993 = 3 \times 331$ | |||||||||||
\(\ds 5993 - 2 \times 51^2\) | \(=\) | \(\ds 791\) | which is composite: $791 = 7 \times 113$ | |||||||||||
\(\ds 5993 - 2 \times 52^2\) | \(=\) | \(\ds 585\) | which is composite: $585 = 3^2 \times 5 \times 13$ | |||||||||||
\(\ds 5993 - 2 \times 53^2\) | \(=\) | \(\ds 375\) | which is composite: $375 = 3 \times 5^3$ | |||||||||||
\(\ds 5993 - 2 \times 54^2\) | \(=\) | \(\ds 161\) | 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
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $5777$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $5777$