# 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:

 $\displaystyle 5993 - 2 \times 0^2$ $=$ $\displaystyle 5993$ 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.