# show that $n\Upsilon_{n-1} \equiv -1 \pmod{n}$ iff $n$ is prime

The Bernoulli numbers $$B_n$$. where all numbers $$B_n$$ are zero with odd index $$n>1$$. first values are given by $$B_{0} = 1$$ , $$B_{1} = -1/2$$, $$B_{2} = 1/6$$, $$B_{3} = -1/30$$.

Agoh conjecture: let $$n$$ be a positive integer with $$n \ge 2$$, then

$$nB_{n-1} \equiv -1 \pmod{n}\iff n\text{ is prime}$$

The idea of ​​$$\Upsilon$$ number comes from my power sum formula and negative values of zeta function

Definition let $$m$$ be a non-negative integer $$\zeta(-m)=(-1)^{m}\frac{B_{m+1}}{m+1}=\sum_{b=1}^{m+1} \Upsilon_b\sum_{i=0}^{b-1} (-1)^{i}(b-i)^{m}\binom{b-1}i$$

so we can calculate values of $$\Upsilon_b$$ , if we substitute value $$\zeta (0)=-1/2$$ gives $$\Upsilon_1=-1/2$$.

Again we can calculate next value using or substituting previous values of $$\Upsilon_b$$.

first values are given by \begin{align*} \Upsilon_1&=-\frac{ 1}{2},\\ \Upsilon_2&=\frac{ 5}{12},\\ \Upsilon_3&=-\frac{ 3}{8},\\ \Upsilon_4&=\frac{ 251}{720},\\ \Upsilon_5&=-\frac{ 95}{288},\\ \Upsilon_6&=\frac{ 19087}{60480},\\ \Upsilon_7&=-\frac {5257}{17280}. \end{align*}

hypothesis: let $$n$$ be a positive integer with $$n \ge 2$$, then

$$n\Upsilon_{n-1} \equiv -1 \pmod{n} \iff n\text{ is prime}$$

Else show that hypothesis is equivalence to agoh's conjectures

Else show that if n is prime then $$n\Upsilon_{n-1} \equiv -1 \pmod{n}$$

Steven Clark had verified conjecture $$n\Upsilon_{n-1} \equiv -1 \pmod{n}\iff n\in \mathbb{P}$$ for $$2 \leq n \leq 500$$

More simply

So $$D_{m,m+1}=m!$$

$$\Upsilon_{m+1}=\frac{(-1)^{m}B_{m+1}-(m+1)(\Upsilon_1 D_{m,1}+\Upsilon_2 D_{m,2}+\cdots+\Upsilon_m D_{m,m})}{(m+1)!}$$

Please see my post on MO

• Are you familiar with Giuga's conjecture? It's in the same territory. en.wikipedia.org/wiki/Agoh–Giuga_conjecture Aug 10, 2019 at 6:48
• @Gerry_Myerson yes I am giuga's and agoh's conjectures are equivalence Aug 10, 2019 at 7:02
• So, what evidence do you have for your hypothesis? Aug 10, 2019 at 7:10
• @Gerry_Myerson Based on given $\Upsilon_n$ values I have assumed my hypothesis is true but I am working on a program for further observation Aug 10, 2019 at 10:41
• I deleted one of my earlier comments after I found an error in my code. Assuming no more errors in my code, I've now verified your conjecture $n\,\Upsilon_{n-1}\equiv -1\,(mod\ n)\,\iff\,n\in \mathbb{P}$ for $2\le n\le 500$. My interest in this question is partially motivated by my conjectured formulas for $\zeta(-j)$ in my question math.stackexchange.com/q/3262873. Aug 26, 2019 at 19:41

Assuming no errors in my code, I have now verified the conjecture $$n\,\Upsilon_{n-1}\equiv -1\,(mod\ n)\,\iff\,n\in \mathbb{P}$$ for $$2\le n\le 1000$$.
I attached an image of my Mathematica code to provide others the opportunity to verify correctness of my code and use it to explore the conjecture for a larger range of $$n$$ values if so desired.