Goldbach's Lesser Conjecture/5777

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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.


Sources