求证题

lihpb00

我记得是用欧拉常数

王守恩

“考研题”！

a,b是正整数,  满足 \(\bigg\lceil\frac{n+a/b}{\sqrt[n]{\pi}}\bigg\rceil\)=n,  n=1,2,3,4,5,...,  这样的{a,b}是怎样的一些数对？

northwolves

王守恩 发表于 2023-11-28 16:25
a,b是正整数,  满足 \(\D\bigg\lceil\frac{a+n}{\sqrt[n]{b}}\bigg\rceil\)=n,  n=1,2,3,4,5,...,  这样的 ...

$a<=n(b^{\frac1n}-1) \rightarrow a<=lnb$

northwolves

王守恩 发表于 2023-11-30 10:44
“考研题”！

a,b是正整数,  满足 \(\bigg\lceil\frac{n+a/b}{\sqrt[n]{\pi}}\bigg\rceil\)=n,  n=1,2,3,4 ...

$a/b<=n(\pi^{\frac1n}-1) \rightarrow ln\pi-1<= a/b<=ln\pi$

王守恩

northwolves 发表于 2023-11-30 12:24
$a/b

“考研题”！
a,b是正整数,  满足 \(\bigg\lceil\frac{n+a/b}{\sqrt[n]{\pi}}\bigg\rceil\)=n,  n=1,2,3,4,5,...,  这样的{a,b}是怎样的一些数对？
什么规律？
a=01, b=01--07,
a=02, b=02--14,
a=03, b=03--21,
a=04, b=04--28,
a=05, b=05--35,
a=06, b=06--42,
a=07, b=07--49,
a=08, b=07--56,
a=09, b=08--63,
a=10, b=09--70,
a=11, b=10--77,
a=12, b=11--84,
a=13, b=12--91,
a=14, b=13--98,
a=15, b=14--105,
a=16, b=14--112,
a=17, b=15--119,
a=18, b=16--126,
a=19, b=17--133,
a=20, b=18--140,
a=21, b=19--147,
a=22, b=20--154,
a=23, b=21--161,
a=24, b=21--,
a=25, b=22--,
a=26, b=23--,
a=27, b=24--,
a=28, b=25--,
a=29, b=26--,
a=30, b=27--,
a=31, b=27--,

northwolves

$\ceil\frac{a}{ln\pi}<= b <= \lfloor\frac{a}{ln\pi-1}\rfloor$

northwolves

王守恩 发表于 2023-11-30 15:50
“考研题”！
a,b是正整数,  满足 \(\bigg\lceil\frac{n+a/b}{\sqrt[n]{\pi}}\bigg\rceil\)=n,  n=1,2,3,4 ...

1. Array[p = Log@Pi; {#, {Ceiling[#/p], Floor[#/(p - 1)]}} &, 100]

复制代码

{{1,{1,6}},{2,{2,13}},{3,{3,20}},{4,{4,27}},{5,{5,34}},{6,{6,41}},{7,{7,48}},{8,{7,55}},{9,{8,62}},{10,{9,69}},{11,{10,76}},{12,{11,82}},{13,{12,89}},{14,{13,96}},{15,{14,103}},{16,{14,110}},{17,{15,117}},{18,{16,124}},{19,{17,131}},{20,{18,138}},{21,{19,145}},{22,{20,152}},{23,{21,158}},{24,{21,165}},{25,{22,172}},{26,{23,179}},{27,{24,186}},{28,{25,193}},{29,{26,200}},{30,{27,207}},{31,{28,214}},{32,{28,221}},{33,{29,228}},{34,{30,234}},{35,{31,241}},{36,{32,248}},{37,{33,255}},{38,{34,262}},{39,{35,269}},{40,{35,276}},{41,{36,283}},{42,{37,290}},{43,{38,297}},{44,{39,304}},{45,{40,310}},{46,{41,317}},{47,{42,324}},{48,{42,331}},{49,{43,338}},{50,{44,345}},{51,{45,352}},{52,{46,359}},{53,{47,366}},{54,{48,373}},{55,{49,380}},{56,{49,386}},{57,{50,393}},{58,{51,400}},{59,{52,407}},{60,{53,414}},{61,{54,421}},{62,{55,428}},{63,{56,435}},{64,{56,442}},{65,{57,449}},{66,{58,456}},{67,{59,462}},{68,{60,469}},{69,{61,476}},{70,{62,483}},{71,{63,490}},{72,{63,497}},{73,{64,504}},{74,{65,511}},{75,{66,518}},{76,{67,525}},{77,{68,532}},{78,{69,538}},{79,{70,545}},{80,{70,552}},{81,{71,559}},{82,{72,566}},{83,{73,573}},{84,{74,580}},{85,{75,587}},{86,{76,594}},{87,{77,601}},{88,{77,608}},{89,{78,614}},{90,{79,621}},{91,{80,628}},{92,{81,635}},{93,{82,642}},{94,{83,649}},{95,{83,656}},{96,{84,663}},{97,{85,670}},{98,{86,677}},{99,{87,684}},{100,{88,690}}}

王守恩

northwolves 发表于 2023-11-30 18:26
{{1,{1,6}},{2,{2,13}},{3,{3,20}},{4,{4,27}},{5,{5,34}},{6,{6,41}},{7,{7,48}},{8,{7,55}},{9,{8,62 ...

还是不会用。EulerGamma=0.5772156649
a,b是正整数,  满足 \(\bigg\lceil\frac{n-a/b}{\sqrt[n]{EulerGamma}}\bigg\rceil\)=n,  n=1,2,3,4,5,...,  这样的{a,b}是怎样的一些数对？
什么规律？
a=01, b=？,
a=02, b=03,
a=03, b=04--5,
a=04, b=05--7,
a=05, b=06--9,
a=06, b=07--10,
a=07, b=08--12,
a=08, b=09--14,
a=09, b=10--16,
a=10, b=11--18,
a=11, b=12--20,
a=12, b=13--21,

northwolves

王守恩 发表于 2023-12-1 10:21
还是不会用。EulerGamma=0.5772156649
a,b是正整数,  满足 \(\bigg\lceil\frac{n-a/b}{\sqrt[n]{EulerGamm ...

1. Array[p=Log@EulerGamma;{#,{Ceiling[#/p],Floor[#/(p-1)]}}&,100]

复制代码

{{1,{-1,-1}},{2,{-3,-2}},{3,{-5,-2}},{4,{-7,-3}},{5,{-9,-4}},{6,{-10,-4}},{7,{-12,-5}},{8,{-14,-6}},{9,{-16,-6}},{10,{-18,-7}},{11,{-20,-8}},{12,{-21,-8}},{13,{-23,-9}},{14,{-25,-10}},{15,{-27,-10}},{16,{-29,-11}},{17,{-30,-11}},{18,{-32,-12}},{19,{-34,-13}},{20,{-36,-13}},{21,{-38,-14}},{22,{-40,-15}},{23,{-41,-15}},{24,{-43,-16}},{25,{-45,-17}},{26,{-47,-17}},{27,{-49,-18}},{28,{-50,-19}},{29,{-52,-19}},{30,{-54,-20}},{31,{-56,-21}},{32,{-58,-21}},{33,{-60,-22}},{34,{-61,-22}},{35,{-63,-23}},{36,{-65,-24}},{37,{-67,-24}},{38,{-69,-25}},{39,{-70,-26}},{40,{-72,-26}},{41,{-74,-27}},{42,{-76,-28}},{43,{-78,-28}},{44,{-80,-29}},{45,{-81,-30}},{46,{-83,-30}},{47,{-85,-31}},{48,{-87,-31}},{49,{-89,-32}},{50,{-90,-33}},{51,{-92,-33}},{52,{-94,-34}},{53,{-96,-35}},{54,{-98,-35}},{55,{-100,-36}},{56,{-101,-37}},{57,{-103,-37}},{58,{-105,-38}},{59,{-107,-39}},{60,{-109,-39}},{61,{-111,-40}},{62,{-112,-41}},{63,{-114,-41}},{64,{-116,-42}},{65,{-118,-42}},{66,{-120,-43}},{67,{-121,-44}},{68,{-123,-44}},{69,{-125,-45}},{70,{-127,-46}},{71,{-129,-46}},{72,{-131,-47}},{73,{-132,-48}},{74,{-134,-48}},{75,{-136,-49}},{76,{-138,-50}},{77,{-140,-50}},{78,{-141,-51}},{79,{-143,-51}},{80,{-145,-52}},{81,{-147,-53}},{82,{-149,-53}},{83,{-151,-54}},{84,{-152,-55}},{85,{-154,-55}},{86,{-156,-56}},{87,{-158,-57}},{88,{-160,-57}},{89,{-161,-58}},{90,{-163,-59}},{91,{-165,-59}},{92,{-167,-60}},{93,{-169,-61}},{94,{-171,-61}},{95,{-172,-62}},{96,{-174,-62}},{97,{-176,-63}},{98,{-178,-64}},{99,{-180,-64}},{100,{-181,-65}}}

王守恩

我就是羡慕您有那么多工具！

有这么一道题:  真分数4/n=1/x+1/y+1/z,   其中n,x,y,z均为正整数。
我们约定x≤y≤z,  且\(\lceil\frac{n+1}{4}\rceil\)=k,  则x的取值范围: k≤x≤3k-1, 这范围太宽了，
编程1: x={k+0,k+1,k+2,k+3,k+4,k+5},  x只要取遍这6个数,  4/n=1/x+1/y+1/z还有哪些n是无解?
编程2: x={k+0,k+1,k+2,k+3,....,k+14},  x只要取遍这15个数,  4/n=1/x+1/y+1/z还有哪些n是无解的?