# Goldbach's Lesser Conjecture/5777

## Goldbach's Lesser Conjecture: $5777$ is a Stern Number

The number $5777$ cannot be represented in the form:

$5777 = 2 a^2 + p$

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 $2 a^2 \le 5777$, it is never the case that $5777 - 2 a^2$ is prime.

Thus:

 $\displaystyle 5777 - 2 \times 0^2$ $=$ $\displaystyle 5777$ which is composite: $5777 = 53 \times 109$ $\displaystyle 5777 - 2 \times 1^2$ $=$ $\displaystyle 5775$ which is composite: $5775 = 3 \times 5^2 \times 7 \times 11$ $\displaystyle 5777 - 2 \times 2^2$ $=$ $\displaystyle 5769$ which is composite: $5769 = 3^2 \times 641$ $\displaystyle 5777 - 2 \times 3^2$ $=$ $\displaystyle 5759$ which is composite: $5759 = 13 \times 443$ $\displaystyle 5777 - 2 \times 4^2$ $=$ $\displaystyle 5745$ which is composite: $5745 = 3 \times 5 \times 383$ $\displaystyle 5777 - 2 \times 5^2$ $=$ $\displaystyle 5727$ which is composite: $5727 = 3 \times 23 \times 83$ $\displaystyle 5777 - 2 \times 6^2$ $=$ $\displaystyle 5705$ which is composite: $5705 = 5 \times 7 \times 163$ $\displaystyle 5777 - 2 \times 7^2$ $=$ $\displaystyle 5679$ which is composite: $5679 = 3^2 \times 631$ $\displaystyle 5777 - 2 \times 8^2$ $=$ $\displaystyle 5649$ which is composite: $5649 = 3 \times 7 \times 269$ $\displaystyle 5777 - 2 \times 9^2$ $=$ $\displaystyle 5615$ which is composite: $5615 = 5 \times 1123$ $\displaystyle 5777 - 2 \times 10^2$ $=$ $\displaystyle 5577$ which is composite: $5577 = 3 \times 11 \times 13^2$ $\displaystyle 5777 - 2 \times 11^2$ $=$ $\displaystyle 5535$ which is composite: $5535 = 3^3 \times 5 \times 41$ $\displaystyle 5777 - 2 \times 12^2$ $=$ $\displaystyle 5489$ which is composite: $5489 = 11 \times 499$ $\displaystyle 5777 - 2 \times 13^2$ $=$ $\displaystyle 5439$ which is composite: $5439 = 3 \times 7 \times 259$ $\displaystyle 5777 - 2 \times 14^2$ $=$ $\displaystyle 5385$ which is composite: $5385 = 3 \times 5 \times 359$ $\displaystyle 5777 - 2 \times 15^2$ $=$ $\displaystyle 5327$ which is composite: $5327 = 7 \times 761$ $\displaystyle 5777 - 2 \times 16^2$ $=$ $\displaystyle 5265$ which is composite: $5265 = 3^4 \times 5 \times 13$ $\displaystyle 5777 - 2 \times 17^2$ $=$ $\displaystyle 5199$ which is composite: $5199 = 3 \times 1733$ $\displaystyle 5777 - 2 \times 18^2$ $=$ $\displaystyle 5129$ which is composite: $5129 = 23 \times 223$ $\displaystyle 5777 - 2 \times 19^2$ $=$ $\displaystyle 5055$ which is composite: $5055 = 3 \times 5 \times 337$ $\displaystyle 5777 - 2 \times 20^2$ $=$ $\displaystyle 4977$ which is composite: $4977 = 3^2 \times 7 \times 79$ $\displaystyle 5777 - 2 \times 21^2$ $=$ $\displaystyle 4895$ which is composite: $4895 = 5 \times 11 \times 89$ $\displaystyle 5777 - 2 \times 22^2$ $=$ $\displaystyle 4809$ which is composite: $4809 = 3 \times 7 \times 229$ $\displaystyle 5777 - 2 \times 23^2$ $=$ $\displaystyle 4719$ which is composite: $4719 = 3 \times 11^2 \times 13$ $\displaystyle 5777 - 2 \times 24^2$ $=$ $\displaystyle 4625$ which is composite: $4625 = 5^3 \times 37$ $\displaystyle 5777 - 2 \times 25^2$ $=$ $\displaystyle 4527$ which is composite: $4527 = 3^2 \times 503$ $\displaystyle 5777 - 2 \times 26^2$ $=$ $\displaystyle 4425$ which is composite: $4425 = 3 \times 5^2 \times 59$ $\displaystyle 5777 - 2 \times 27^2$ $=$ $\displaystyle 4319$ which is composite: $4319 = 7 \times 617$ $\displaystyle 5777 - 2 \times 28^2$ $=$ $\displaystyle 4209$ which is composite: $4209 = 3 \times 23 \times 61$ $\displaystyle 5777 - 2 \times 29^2$ $=$ $\displaystyle 4095$ which is composite: $4095 = 3^2 \times 5 \times 7 \times 13$ $\displaystyle 5777 - 2 \times 30^2$ $=$ $\displaystyle 3977$ which is composite: $3977 = 41 \times 97$ $\displaystyle 5777 - 2 \times 31^2$ $=$ $\displaystyle 3855$ which is composite: $3855 = 3 \times 5 \times 257$ $\displaystyle 5777 - 2 \times 32^2$ $=$ $\displaystyle 3729$ which is composite: $3729 = 3 \times 11 \times 113$ $\displaystyle 5777 - 2 \times 33^2$ $=$ $\displaystyle 3599$ which is composite: $3599 = 60^2 - 1 = \left({60 + 1}\right) \left({60 - 1}\right) = 59 \times 61$ $\displaystyle 5777 - 2 \times 34^2$ $=$ $\displaystyle 3465$ which is composite: $3465 = 3^2 \times 5 \times 7 \times 11$ $\displaystyle 5777 - 2 \times 35^2$ $=$ $\displaystyle 3327$ which is composite: $3327 = 3 \times 1109$ $\displaystyle 5777 - 2 \times 36^2$ $=$ $\displaystyle 3185$ which is composite: $3185 = 5 \times 7^2 \times 13$ $\displaystyle 5777 - 2 \times 37^2$ $=$ $\displaystyle 3039$ which is composite: $3039 = 3 \times 1013$ $\displaystyle 5777 - 2 \times 38^2$ $=$ $\displaystyle 2889$ which is composite: $2889 = 3^3 \times 107$ $\displaystyle 5777 - 2 \times 39^2$ $=$ $\displaystyle 2735$ which is composite: $2735 = 5 \times 547$ $\displaystyle 5777 - 2 \times 40^2$ $=$ $\displaystyle 2577$ which is composite: $2577 = 3 \times 859$ $\displaystyle 5777 - 2 \times 41^2$ $=$ $\displaystyle 2415$ which is composite: $2415 = 3 \times 5 \times 7 \times 23$ $\displaystyle 5777 - 2 \times 42^2$ $=$ $\displaystyle 2249$ which is composite: $2249 = 13 \times 173$ $\displaystyle 5777 - 2 \times 43^2$ $=$ $\displaystyle 2079$ which is composite: $2079 = 3^3 \times 7 \times 11$ $\displaystyle 5777 - 2 \times 44^2$ $=$ $\displaystyle 1905$ which is composite: $1905 = 3 \times 5 \times 127$ $\displaystyle 5777 - 2 \times 45^2$ $=$ $\displaystyle 1727$ which is composite: $1727 = 11 \times 157$ $\displaystyle 5777 - 2 \times 46^2$ $=$ $\displaystyle 1545$ which is composite: $1545 = 3 \times 5 \times 103$ $\displaystyle 5777 - 2 \times 47^2$ $=$ $\displaystyle 1359$ which is composite: $1359 = 3^2 \times 151$ $\displaystyle 5777 - 2 \times 48^2$ $=$ $\displaystyle 1169$ which is composite: $1169 = 7 \times 167$ $\displaystyle 5777 - 2 \times 49^2$ $=$ $\displaystyle 975$ which is composite: $975 = 3 \times 5^2 \times 13$ $\displaystyle 5777 - 2 \times 50^2$ $=$ $\displaystyle 777$ which is composite: $777 = 3 \times 7 \times 37$ $\displaystyle 5777 - 2 \times 51^2$ $=$ $\displaystyle 575$ which is composite: $575 = 5^2 \times 23$ $\displaystyle 5777 - 2 \times 52^2$ $=$ $\displaystyle 369$ which is composite: $369 = 3^2 \times 41$ $\displaystyle 5777 - 2 \times 53^2$ $=$ $\displaystyle 159$ which is composite: $159 = 3 \times 53$

That exhausts all $a$ such that $2 a^2 \le 5775$, as $2 \times 54^2 = 5832$.

Hence the result.

$\blacksquare$

## Source of Name

This entry was named for Christian Goldbach.