A Note on the Existence of Certain Infinite Families of Imaginary Quadratic Fields

Let D<0 be the fundamental discriminant of an imaginary quadratic field, and h(D) its class number. In this paper, we show that for any prime p>3 and &epsiv;=−1,0, or 1,<f>#<fen><cp type="lcub" style="s">−X<D<0∣h(D)≢0(mod p) and<fen><cp type="lpar" style="s">D/p</fr><cp type="rpar" style="s"></fen>=&epsiv;<cp type="rcub" style="s"></fen>≫p<fr SHAPE="BUILT" ALIGN="C" STYLE="S"><rad><rcd>X</rcd></rad>/log X<cp type="rpar" style="s"></fen>=&epsiv;<cp type="rcub" style="s"></fen>≫p<rad><rcd>X</rcd></rad>/log X</f> [Copyright &y& Elsevier]