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:

$\ds 5777 - 2 \times 0^2$

$=$

$\ds 5777$

which is composite: $5777 = 53 \times 109$

$\ds 5777 - 2 \times 1^2$

$=$

$\ds 5775$

which is composite: $5775 = 3 \times 5^2 \times 7 \times 11$

$\ds 5777 - 2 \times 2^2$

$=$

$\ds 5769$

which is composite: $5769 = 3^2 \times 641$

$\ds 5777 - 2 \times 3^2$

$=$

$\ds 5759$

which is composite: $5759 = 13 \times 443$

$\ds 5777 - 2 \times 4^2$

$=$

$\ds 5745$

which is composite: $5745 = 3 \times 5 \times 383$

$\ds 5777 - 2 \times 5^2$

$=$

$\ds 5727$

which is composite: $5727 = 3 \times 23 \times 83$

$\ds 5777 - 2 \times 6^2$

$=$

$\ds 5705$

which is composite: $5705 = 5 \times 7 \times 163$

$\ds 5777 - 2 \times 7^2$

$=$

$\ds 5679$

which is composite: $5679 = 3^2 \times 631$

$\ds 5777 - 2 \times 8^2$

$=$

$\ds 5649$

which is composite: $5649 = 3 \times 7 \times 269$

$\ds 5777 - 2 \times 9^2$

$=$

$\ds 5615$

which is composite: $5615 = 5 \times 1123$

$\ds 5777 - 2 \times 10^2$

$=$

$\ds 5577$

which is composite: $5577 = 3 \times 11 \times 13^2$

$\ds 5777 - 2 \times 11^2$

$=$

$\ds 5535$

which is composite: $5535 = 3^3 \times 5 \times 41$

$\ds 5777 - 2 \times 12^2$

$=$

$\ds 5489$

which is composite: $5489 = 11 \times 499$

$\ds 5777 - 2 \times 13^2$

$=$

$\ds 5439$

which is composite: $5439 = 3 \times 7 \times 259$

$\ds 5777 - 2 \times 14^2$

$=$

$\ds 5385$

which is composite: $5385 = 3 \times 5 \times 359$

$\ds 5777 - 2 \times 15^2$

$=$

$\ds 5327$

which is composite: $5327 = 7 \times 761$

$\ds 5777 - 2 \times 16^2$

$=$

$\ds 5265$

which is composite: $5265 = 3^4 \times 5 \times 13$

$\ds 5777 - 2 \times 17^2$

$=$

$\ds 5199$

which is composite: $5199 = 3 \times 1733$

$\ds 5777 - 2 \times 18^2$

$=$

$\ds 5129$

which is composite: $5129 = 23 \times 223$

$\ds 5777 - 2 \times 19^2$

$=$

$\ds 5055$

which is composite: $5055 = 3 \times 5 \times 337$

$\ds 5777 - 2 \times 20^2$

$=$

$\ds 4977$

which is composite: $4977 = 3^2 \times 7 \times 79$

$\ds 5777 - 2 \times 21^2$

$=$

$\ds 4895$

which is composite: $4895 = 5 \times 11 \times 89$

$\ds 5777 - 2 \times 22^2$

$=$

$\ds 4809$

which is composite: $4809 = 3 \times 7 \times 229$

$\ds 5777 - 2 \times 23^2$

$=$

$\ds 4719$

which is composite: $4719 = 3 \times 11^2 \times 13$

$\ds 5777 - 2 \times 24^2$

$=$

$\ds 4625$

which is composite: $4625 = 5^3 \times 37$

$\ds 5777 - 2 \times 25^2$

$=$

$\ds 4527$

which is composite: $4527 = 3^2 \times 503$

$\ds 5777 - 2 \times 26^2$

$=$

$\ds 4425$

which is composite: $4425 = 3 \times 5^2 \times 59$

$\ds 5777 - 2 \times 27^2$

$=$

$\ds 4319$

which is composite: $4319 = 7 \times 617$

$\ds 5777 - 2 \times 28^2$

$=$

$\ds 4209$

which is composite: $4209 = 3 \times 23 \times 61$

$\ds 5777 - 2 \times 29^2$

$=$

$\ds 4095$

which is composite: $4095 = 3^2 \times 5 \times 7 \times 13$

$\ds 5777 - 2 \times 30^2$

$=$

$\ds 3977$

which is composite: $3977 = 41 \times 97$

$\ds 5777 - 2 \times 31^2$

$=$

$\ds 3855$

which is composite: $3855 = 3 \times 5 \times 257$

$\ds 5777 - 2 \times 32^2$

$=$

$\ds 3729$

which is composite: $3729 = 3 \times 11 \times 113$

$\ds 5777 - 2 \times 33^2$

$=$

$\ds 3599$

which is composite: $3599 = 60^2 - 1 = \left({60 + 1}\right) \left({60 - 1}\right) = 59 \times 61$

$\ds 5777 - 2 \times 34^2$

$=$

$\ds 3465$

which is composite: $3465 = 3^2 \times 5 \times 7 \times 11$

$\ds 5777 - 2 \times 35^2$

$=$

$\ds 3327$

which is composite: $3327 = 3 \times 1109$

$\ds 5777 - 2 \times 36^2$

$=$

$\ds 3185$

which is composite: $3185 = 5 \times 7^2 \times 13$

$\ds 5777 - 2 \times 37^2$

$=$

$\ds 3039$

which is composite: $3039 = 3 \times 1013$

$\ds 5777 - 2 \times 38^2$

$=$

$\ds 2889$

which is composite: $2889 = 3^3 \times 107$

$\ds 5777 - 2 \times 39^2$

$=$

$\ds 2735$

which is composite: $2735 = 5 \times 547$

$\ds 5777 - 2 \times 40^2$

$=$

$\ds 2577$

which is composite: $2577 = 3 \times 859$

$\ds 5777 - 2 \times 41^2$

$=$

$\ds 2415$

which is composite: $2415 = 3 \times 5 \times 7 \times 23$

$\ds 5777 - 2 \times 42^2$

$=$

$\ds 2249$

which is composite: $2249 = 13 \times 173$

$\ds 5777 - 2 \times 43^2$

$=$

$\ds 2079$

which is composite: $2079 = 3^3 \times 7 \times 11$

$\ds 5777 - 2 \times 44^2$

$=$

$\ds 1905$

which is composite: $1905 = 3 \times 5 \times 127$

$\ds 5777 - 2 \times 45^2$

$=$

$\ds 1727$

which is composite: $1727 = 11 \times 157$

$\ds 5777 - 2 \times 46^2$

$=$

$\ds 1545$

which is composite: $1545 = 3 \times 5 \times 103$

$\ds 5777 - 2 \times 47^2$

$=$

$\ds 1359$

which is composite: $1359 = 3^2 \times 151$

$\ds 5777 - 2 \times 48^2$

$=$

$\ds 1169$

which is composite: $1169 = 7 \times 167$

$\ds 5777 - 2 \times 49^2$

$=$

$\ds 975$

which is composite: $975 = 3 \times 5^2 \times 13$

$\ds 5777 - 2 \times 50^2$

$=$

$\ds 777$

which is composite: $777 = 3 \times 7 \times 37$

$\ds 5777 - 2 \times 51^2$

$=$

$\ds 575$

which is composite: $575 = 5^2 \times 23$

$\ds 5777 - 2 \times 52^2$

$=$

$\ds 369$

which is composite: $369 = 3^2 \times 41$

$\ds 5777 - 2 \times 53^2$

$=$

$\ds 159$

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.

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/5777

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