Fu XudanFermat's Last Theorem states that x~n+y~n=z~nhas no positive integer solutions for x, y and z when n＞2. In order to prove Fermat's LastTheorem one need only show that no fourth power is a sum of two fourth powers, and for allodd prime p, no p-th power is a sum of two p-th powers. In one of the few actual proofs heleft behind, Fermat himself had dealt with case n=4. And when n is odd prime p, Fermat'sLast Theorem could be broken into two cases:CaseⅠ: x~p+y~P=z~p has no positive integer solutions for which x, y and z are relativelyprime to p;CaseⅡ: x~p+y~p=z~p has no positive integer solutions for which one and only one ofthe three numbers is divisible by p.Wieferich[24] discovered the following criterion:If p(?)q_2(p), where q_m(p)=(m~(p-1)-1)/pis the Fermat quotient, then CaseⅠis true. Mirimanoff [19] proved this for q_3(P). In [17],Lehmer mentioned that q_2(p)≡q_3(p)≡0(mod p)(?) sum from r=1 to [p/n] 1/r≡0(mod p), n=2,3,4,6.Combining the results of Wieferich, Mirimanoff and Lehmer, we could describe as follows: If sum from r=1 to [p/n] 1/r(?)0(mod p), for n=2,3,4 or 6, then CaseⅠis true. Thus, Lehmer's congruence sum from r=1 to (p-1)/2 1/r≡-2q_2(p)+pq_2~2(p) (mod p~2), (*)where p is odd prime, plays an important role in proving Fermat's Last Theorem. And it'smeaningful to study the congruences as above.The content of this paper is divided into three parts.In the first part, we generalize Lehmer's congruences to arbitrary positive integer mod-uli, e.g., sum from r=1,(r,n)=1 to [n/2] 1/(n-2r)≡q_2(n)-1/2 nq_2~2(n) (mod n~2), if (6, n)=1.As an application, we obtain several congruences of binomial coefficients concerning gener-alized Euler totient function. Meanwhile, Lehmer's another congruence sum from r=1 to [p/4] 1/r~2≡(-1)~((p-1)/2)4E_(p-3) (mod p)is also generalized to arbitrary prime power moduli.In the second part, inspired by Zhao's work, we study the natural generalization of hisresult: S(1,1,1;p)≡-2B_(p-3) (mod p).And we obtain the explicit expression of S(α,β,γ; p), whereα,β,γare positive integers. Asan application, we derive some congruences concerning with Catalan numbers.In the third part, we consider binomial coefficient congruences of Babbage, Glaisherand Ljunggren and derive their corresponding u-nomial coefficient congruences respectively,which partly generalize Andrews' results, meanwhile we obtain some properties of Fibonaccinumbers. 0.1 Congruences involving the quotients of Euler and its applicationIn 1938, Lehmer[17] established a famous congruence sum from r=1 to (p-1)/2 1/r≡-2q_2(p)+pq_2~2(p) (mod p~2),where q_2(p)=(2~(φ(p))-1)/pis Euler's quotient, which, along with some other congruences, plays an important role inproving the first case of Fermat's Last Theorem. In 2002, Cai[3] showed that for any oddn＞1, (?)1/r≡-2q_2(n)+nq_2~2(n) (mod n~2),where q_i(n)=i~(φ(n))-1/n, (i,n)=1is Euler's quotient of n with base i. In this section, we wish to generalize some othercongruences of Lehmer to arbitrary positive integer moduli: sum from r=1 to (p-1)/2 1/(p-2r)≡q_2(p)-pq_2~2(p) (mod p~2), (0.1.1) sum from r=1 to [p/3] 1/(p-3r)≡1/2 q_3(p)-1/4 pq_3~2(p) (mod p~2), p＞3 (0.1.2) sum from r=1 to [p/4] 1/(p-4r)≡3/4 q_2(p)-3/8 pq_2~2(p) (mod p~2), p＞3 (0.1.3) sum from r=1 to [p/6] 1/(p-6r)≡1/3 q_2(p)+1/4 q_3(p)-1/6 pq_2~2(p)-1/8 pq_3~2(p) (mod p~2), p＞5 (0.1.4)where [x] denotes the intern part of x.In the first part, we generalize (0.1.1)-(0.1.4) and get the following. Theorem 1 If n is odd, then sum from r=1,(r,n)=1 to [n/2] 1/(n-2r)≡q_2(n)-1/2 nq_2~2(n) (mod n~2), (3, n)=1 sum from r=1,(r,n)=1 to [n/3] 1/(n-3r)≡1/2 q_3(n)-1/4 nq_3~2(n) (mod n~2), (3,n)=1 sum from r=1,(r,n)=1 to [n/4] 1/(n-4r)≡3/4 q_2(n)-3/8 nq_2~2(n) (mod n~2), (3,n)=1 sum from r=1,(r,n)=1 to [n/6] 1/(n-6r)≡1/3 q_2(n)+1/4 q_3(n)-1/6 nq_2~2(n)-1/8 nq_3~2(n) (mod n~2), (15,n)=1As an application of Theorem 1, we have the following result.Theorem 2 If n is odd, then multiply from d/n (?)~(μ(n/d))≡(-1)~(φ_3(n))(3~(φ(n)+1)-1)/2 (mod n~2), (3,n)=1, (0.1.5) multiply from d/n (?)~(μ(n/d))≡(-1)~(φ_4(n))(3·2~(φ(n))-2) (mod n~2), (3,n)=1, (0.1.6) multiply from d/n (?)~(μ(n/d))≡(-1)~(φ_6(n))(2~(φ(n)+2)+3~(φ(n)+1)-5)/2 (mod n~2), (15,n)=1, (0.1.7)whereφ_e(n)=integral from d/nμ(n/d)[d/e]is the generalized Euler totient function. Remark: We shall prove in Lemma 1.3.1 below that (-1)~(φ_e(n))=-1 only when n isa prime power p~αwith p≡2 (mod 3) for (0.1.5), p≡5 or 7 (mod 8) for (0.1.6) and p≡7or 11 (mod 12) for (0.1.7).Accordingly, we obtain thatTheorem 3 If p, q are distinct odd primes, thenHowever, we have not been able to generalize the following formula of Lehmer to arbi-trary positive integer moduli: integral from r=1 to [p/4] 1/r~2≡(-1)~((p-1)/2)4E_(p-3)(mod p), (0.1.8)for any prime p≥5, where E_(2n) is the 2n-th Euler number which can be defined by thegenerating function: sec x=integral from n=0 to∞(-1)~n E_(2n) x~(2n)/(2n)!, |x|＜π/2.What we prove is the following result.Theorem 4 If p is an odd prime, l≥1, then integral from r=1,(p,r)=1 to [p~l/4] 1/r~2≡(-1)~((p~l-1)/2)4E_(φ(p~l)-2)(mod p~l), In particular, when p=3 we have sum from r=1,(3,r)=1 to [(3~1)/4] 1/(r~2)≡(-1)~((3~1-1)/2)4E_(φ(3~1)-2)(mod 3~(l-1)).0.2 Congruences for finite triple harmonic sumsIn the second section, we study the finite triple harmonic sums concerning Bernoullinumbers, which are concerned a lot in Number Theory. Let us recall some standard notationsand definitions.Bernoulli numbers B_k are defined by the recurrence relation[10] sum from k-0 to n (?)B_k=0(n≥1), B_0=1.It is well-known that B_(2k+1)=0 for k≥1, and B_0=1, B_1=-1/2, B_2=1/6, B_4=-1/30…and so on. Several authors have studied the congruences concerning Bernoulli numbers. Usingpartial sum of multiple zeta value series, Zhao[26] first established the congruenceZhou and Cai[29] generalized it toInspired by their work, we define and study the following finite triple harmonic sums: where n≥3 andα,β,γare positive integers. We refer to w =α+β+γas the weight ofS(α,β,γ;n). In this part, we consider n=p be prime, and assumeα≤β≤γsince theyare symmetric. By dealing finite triple harmonic sums with some elementary techniques, i.e.,the properties of Bernoulli numbers and the recursive method, we get the result as follows.Theorem 5 Let p be prime.(ⅰ) If w is even andp≥w+3, then S(α,β,γ;p)≡0 (mod p(ⅱ) If w is and and p≥w, then S(α,β,γ;p)≡r(α,β,γ)B_(p-w) (mod p), (0.2.2)where r(α,β,γ)=1/w{(-1)~αsum from i-0 toβ-1 (?)(?)+(-1)~βsum from i-0 toα-1 (?)(?)}.By (0.2.2) and the equation sum from i=0 to q (?)(?)=(?),we can derive the following two noteworthy general congruences:Theorem 6 Let n be any positive integer, then S(2n-1,2n, 2n;p)≡0 (rood pfor p≥6n-1 and for p≥6n+1, S(2n, 2n, 2n+1;p)≡C_(2n)B_(p-6n-1) (rood p),where C_k=1/(k+1)(?) is the k-th Catalan number.Now we wish to obtain S(α,β,γ; p) modulo p~2 when w is even. Denote T(α,β,γ;p)= sum from 1≤i+j≤p-1 1/(i~αj~β(i+j)~γ)It's easy to see that T(α,β,γ;p)≡(-1)~γS(α,β,γ;p) (mod p)indeed, we have (-1)~γS(α,β,γ;p)≡T(α,β,γ;p)+γT(α,β,γ+1;p)p (mod p~2) (0.2.3)Definewe have:Theorem 7: when w is even, T(α,β,γ)≡{R(α,β,γ)-w((?))/2(w+1)}B_(p-w-1p) (mod p~2). (0.2.4)By (0.2.4) and (0.2.3), we can derive S(α,β,γ;p) modulo p~2 easily.0.3 Some u-nomial coefficient congruencesMany authors provided q-analogs of some famous congruences, i.e., Andrews[I]. Here weconsider the congruences concerning the generalized binomial coefficient. For fixed nonzerointegers A, B, Lucas sequence {u_n}_(n∈N) is defined by u_0=0, u_1=1,u_(n+1)=Au_n-Bu_(n-1), for n=1, 2,…. It's companion sequence {v_n}_(n∈N)is given as follows v_0=2,v_1=A, v_(n+1)=Av_n-Bv_(n-1),for n=1, 2,…. By induction, we may prove u_n=∑_(0≤k＜n)α~kβ~(n-1-k), whereα=(A+△~(1/2))/2,β=(A-△~(1/2))/2,△A~2-4B.Setting [n]=Π_(0＜k≤n)u_k, Hu and Sun[11] considered Lucas u-nomial coefficient (gener-alized binomial coefficient):In particular, if A=2 and B=1, it is exactly ordinary binomial coefficient ((?) and ifA=q+1, B=q, where q∈Z and |q|＞1, it is Gaussian q-binomial coefficient ((?))_q sinceu_j=(q~j-1)/(q-1) for j=0,1,2,…,i.e.,In the case A=1, B=-1, [n, k] becomes Fibonacci coefficient[25]because u_j=F_j(j=0, 1, 2,…).In 1819, Babbage proved that:for p≥3. In 1900, Glaisher proved that if p≥5, where h＞2 is an integer. In a joint paper of 1952, Ljunggren generalized (0.3.2) toand Jacobsthal tofor any integers h＞j＞0 and p≥5, where a is the power of p dividing p~3hj(h-j). By(0.3.4), (0.3.3) becomesfor integers h＞j＞0 and p≥5.In this paper, using the viewpoint of Hu and Sun, we prove the similar results of(0.3.1)、(0.3.2)、(0.3.5), and get the corresponding congruences on Fibonacci coefficient.For a, b∈Z, let (a, b) denote the greatest common divisor of a and b. We obtain the following:Theorem 8 Let A, B be two nonzero integers satisfying (A, B)=1 and u_p≠±1 bedefined as above, then [2p-1,p-1]≡B~(p(p-1)/2) (mod u_p~2). (0.3.6)Using (0.3.6), we can getTheorem 9 If A, B, u_p are defined as above, then sum from i=1 to p-1 u_(i-1)/u_i≡(p-1)/2 A/B (mod u_p). (0.3.7) Remark: Let A=q+1,B=q, u_i is [i]_q=(1-q~i)/(1-q), then u_(i-1)/u_i=(q~(i-1)-1)/(q~i-1)=1/q(1-1/[i]_q).Therefore, (0.3.7) partly turns out to be Andrews' Theorem 4[1], namely, sum from i=1 to p-1 1/[i]_q≡(p-1)/2(1-q) (mod [p]_q),for odd prime p.Naturally, we continue to consider the u-nomial coefficient congruence of (0.3.2), (0.3.5)and actually derive that:Theorem 10 Let h, B are positive integers, then [hp-1,p-1]≡B~(p(h-1)(p-1)/2) (mod u_p~2). (0.3.8)Let u′_0=0, u′_1=1 and u′_(n+1)=v_qu′_n-B~qu′_(n-1)(n=1, 2,….), denote the u-nomialcoefficient which defined in Lucas sequence {u′_n} by [n, k]′, thenTheorem 11 Let h＞j＞0, B＞0 are integers, then [hp-1,jp-1]≡[h-1,j-1]′B~(pj(h-j)(p-1)/2)(mod u_p~2). (0.3.9)In fact, (0.3.8), (0.3.9) partly generalize Andrews' Theorem 2 and Theorem 3 in [1] re-spectively. Taking A=1, B=-1 in (0.3.6)-(0.3.9), we could derive a series of congruencesconcerning with Fibonacci numbers which are mentioned in the article.