Goldbach's Lesser Conjecture/5993

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): $5777$

Goldbach's Lesser Conjecture/5993

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