Supercongruence And Sums Of Four Polygonal Numbers

Let m>2 and q>0 be integers with m even or q odd.We show the supercon-gruence for any prime p>mq.This confirms a conjecture of Sun.Let m ? 3 be an integer.The polygonal numbers of order m + 2 are given by pm-2(n)?m(2n)+n(n?0 1,2,...).A famous claim of Fermat proved by Cauchy asserts that each nonnegative integer is the sum of m + 2 polygonal numbers of order m + 2.For(a,b)=(1,1),(2,2),(1,3),(2,4),we study whether any sufficiently large integer can be expressed as pm+2(x1)+pm+2(x2)+ apm+2(x3)+ bpm+2(x4)with x1,x2,x3,x4 nonnegative integers.We show that the answer is positive if(a,b)?{(1,3),(2,4)},or(a,b)=(1,1)&4 | m,or(a,6)=(2,2)&m(?)2(mod 4).We also confirm a conjecture of Z.-W.Sun which states that any natural number can be written as p6(x1)+ P6(x2)+ 2P6(x3)+ 4P6(x4)with x1,x2,x3,x4 nonnegative integers.Our main results of supercongruences and sums of polygonal numbers were published in Finite Fields and Their Applications and Acta Arithmatica,respectively.