正整數a,b,c 求a-b最小值

a > b, a = b + k, k = a - b 3^a – 3^b = 3^(b+k) – 3^b = 3^b(3^k – 1) 1000｜3^k – 110｜3^k – 13^k = 10P + 1, k = 4m 1000｜3^(4m) – 1 = 81^m -1 = (81 – 1)(81^(m-1) + … + 81 + 1)1000｜80(81^(m-1) + … + 81 + 1) 25｜2(81^(m-1) + … + 81 + 1)5｜(1 + 81 + 81^2 + 81^3 + 81^4) + (81^5 + … + 81^9) + …m – 1 = 4, 9, 14, … = 5n – 1m = 5n 1000｜81^m -1 = 81^(5n) – 1 = 3486784401^n – 1 = (3486784000 + 401)^n – 11000｜401^n – 1 1000｜401^5 – 1 = 10368641602001 – 1 最小 n = 5K = 4m = 20n = 100 a – b 最小值為100

正整數a，b，c若3^a－3^b=1000c，求a－b最小值=?
Sol
設p=a－b
3^a－3^b
=3^(b+p)－3^b
=3^b*(3^p－1)
1000｜(3^p－1)
3^1=3
3^2=9
3^3=27
3^4=81
3^8=6561
3^12=531441
3^12=>441
3^16=>441*81=357213^16=>7213^20=>721*81=58401
3^20=>401
3^24=>401*81=>32481
3^24=>481
3^28=>481*81=38961
3^28=>961
3^32=>961*81=77841
3^32=>841
3^36=>841*81=68121
3^36=>121
3^40=>121*81=9801
3^40=>801
3^44=>801*81=64881
3^44=>881
3^48=>881*81=71361
3^48=>361
3^52=>361*81=29241
3^52=241
3^56=>241*81=19521
3^56=>521
3^60=>521*81=42201
3^60=>201
3^64=>201*81=16281
3^64=>281
3^68=>281*81=22761
3^68=>761
3^72=>761*81=61641
3^72=>641
3^76=>641*81=51921
3^76=>921
3^80=>921*81=74601
3^80=>601
3^84=>601*81=48681
3^84=>681
3^88=>681*81=55161
3^92=>161*81=13041
3^92=>41
3^96=>41*81=3321
3^96=>321
3^100=>321*81=26001
最少p=100

400以內，剛剛粗略用計算機按了下，應該是100，不過沒啥好的想法= =

2014-10-21 16:52:37 補充：
由題目知a>b，所以3^b[3^(a-b)-1]=1000c
所以1000整除3^b[3^(a-b)-1]，故1000整除3^(a-b)-1
(因為1000與3^b互質)
即解出3^(a-b)≡1(mod 1000)
由尤拉定理，3^(400)≡1(mod 1000)
所以應為400的因數
剛剛試了答案應為100，不過不曉得有啥好想法= =