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Proof of a recent conjecture of Z.-W. Sun | Song Guo
; Victor J. W. Guo
; | Date: |
18 Apr 2016 | Abstract: | The polynomials $d_n(x)$ are defined by egin{align*} d_n(x) &=
sum_{k=0}^n{nchoose k}{xchoose k}2^k. end{align*} We prove that, for any
prime $p$, the following congruences hold modulo $p$: egin{align*}
sum_{k=0}^{p-1}frac{{2kchoose k}}{4^k} d_kleft(-frac{1}{4}
ight)^2
&equiv egin{cases} 2(-1)^{frac{p-1}{4}}x,& ext{if $p=x^2+y^2$ with
$xequiv 1pmod{4}$,} 0,& ext{if $pequiv 3pmod{4}$,} end{cases} [5pt]
sum_{k=0}^{p-1}frac{{2kchoose k}}{4^k} d_kleft(-frac{1}{6}
ight)^2
&equiv 0, quad ext{if $p>3$,} [5pt] sum_{k=0}^{p-1}frac{{2kchoose
k}}{4^k} d_kleft(frac{1}{4}
ight)^2 &equiv egin{cases} 0,& ext{if
$pequiv 1pmod{4}$,} (-1)^{frac{p+1}{4}}{frac{p-1}{2}choose
frac{p-3}{4}},& ext{if $pequiv 3pmod{4}$.} end{cases}
sum_{k=0}^{p-1}frac{{2kchoose k}}{4^k} d_kleft(frac{1}{6}
ight)^2 &equiv
0, quad ext{if $p>5$.} end{align*} The $pequiv 3pmod{4}$ case of the first
one confirms a conjecture of Z.-W. Sun, while the second one confirms a special
case of another conjecture of Z.-W. Sun. | Source: | arXiv, 1604.5019 | Services: | Forum | Review | PDF | Favorites |
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