# Goldbach's Lesser Conjecture/5777

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## Contents

## 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

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $5777$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $5777$