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Number TheoryQuestion and Answers: Page 1

Question Number 222974 Answers: 1 Comments: 0

Question Number 222635 Answers: 0 Comments: 0

Question Number 222516 Answers: 2 Comments: 0

Given the integer k,how to find the incomplete general solution for the non-trivial integer solutions of the Diophantine equation: a^4 +b^4 +ka^2 b^2 =c^4 +d^4 +kc^2 d^2 ,a,b,c,d∈N,k∈Z,gcd(a,b,c,d)=1

$$\mathrm{Given}\:\mathrm{the}\:\mathrm{integer}\:{k},\mathrm{how}\:\:\mathrm{to} \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{incomplete}\:\mathrm{general} \\ $$$$\mathrm{solution}\:\mathrm{for}\:\mathrm{the}\:\mathrm{non}-\mathrm{trivial}\:\mathrm{integer} \\ $$$$\mathrm{solutions}\:\mathrm{of}\:\:\mathrm{the}\:\mathrm{Diophantine}\:\mathrm{equation}: \\ $$$${a}^{\mathrm{4}} +{b}^{\mathrm{4}} +{ka}^{\mathrm{2}} {b}^{\mathrm{2}} ={c}^{\mathrm{4}} +{d}^{\mathrm{4}} +{kc}^{\mathrm{2}} {d}^{\mathrm{2}} ,{a},{b},{c},{d}\in\mathbb{N},{k}\in\mathbb{Z},\mathrm{gcd}\left({a},{b},{c},{d}\right)=\mathrm{1} \\ $$

Question Number 219732 Answers: 1 Comments: 0

Question Number 219657 Answers: 1 Comments: 0

prove; Π_(n=1) ^∞ (((5n−2)(5n−3))/((5n−1)(5n−4))) = ϕ

$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:{prove}; \\ $$$$\:\:\:\:\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\prod}}\:\frac{\left(\mathrm{5}{n}−\mathrm{2}\right)\left(\mathrm{5}{n}−\mathrm{3}\right)}{\left(\mathrm{5}{n}−\mathrm{1}\right)\left(\mathrm{5}{n}−\mathrm{4}\right)}\:=\:\varphi \\ $$$$ \\ $$

Question Number 218438 Answers: 1 Comments: 0

An amazing thing i saw S = 1 + 2 + 3 + 4 + 5 + 6... = 1 + 2(2/2 + 3/2 + 4/2 + 5/2 +6/2....) = 1 + 2(1 + 3/2 + 2 + 5/2 + 3...) = 1 + 2(1+ 2 + 3 ... + 3/2 + 5/2...) = 1 + 2S + 2Σ_(n= 1) ^∞ ((2n + 1)/2) or,S − 2S = 1 + Σ_(n=1) ^∞ 2n + 1 ∴ −S = Σ_(n=0) ^∞ 2n + 1 Sum of all odd numbers! I know the step S−2S = −S is not allowed

Question Number 218208 Answers: 1 Comments: 1

This question is really important Prove or disprove that lim_(n→∞) ((3^n m+3^(n−1) )/2^(⌈(n/2)⌉) ) + (3^(n−1) /2^n ) the limit exists for m ∈ N \B where B = {n ∣ log_2 (n) ∈ N }

$${This}\:{question}\:{is}\:{really}\:{important} \\ $$$${Prove}\:{or}\:{disprove}\:{that} \\ $$$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\frac{\mathrm{3}^{{n}} {m}+\mathrm{3}^{{n}−\mathrm{1}} }{\mathrm{2}^{\lceil\frac{{n}}{\mathrm{2}}\rceil} }\:+\:\frac{\mathrm{3}^{{n}−\mathrm{1}} }{\mathrm{2}^{{n}} }\: \\ $$$$\:{the}\:{limit}\:{exists}\:{for}\:{m}\:\in\:{N}\:\backslash{B} \\ $$$${where}\:{B}\:=\:\left\{{n}\:\mid\:{log}_{\mathrm{2}} \left({n}\right)\:\in\:{N}\:\right\} \\ $$

Question Number 218076 Answers: 2 Comments: 0

How many ways to arrnge the letters ABCCCDEFG (1) in general . (2) all 3 Cs must be together (3) only 2 Cs must be together (4) no 2 or 3 Cs be together (5) no letter still in its original place .

$${How}\:{many}\:{ways}\:{to}\:{arrnge} \\ $$$${the}\:{letters}\:{ABCCCDEFG} \\ $$$$\left(\mathrm{1}\right)\:{in}\:{general}\:. \\ $$$$\left(\mathrm{2}\right)\:{all}\:\mathrm{3}\:{Cs}\:{must}\:{be}\:{together} \\ $$$$\left(\mathrm{3}\right)\:{only}\:\mathrm{2}\:{Cs}\:{must}\:{be}\:{together} \\ $$$$\left(\mathrm{4}\right)\:{no}\:\mathrm{2}\:{or}\:\mathrm{3}\:{Cs}\:{be}\:{together} \\ $$$$\left(\mathrm{5}\right)\:{no}\:{letter}\:{still}\:\:{in}\:{its} \\ $$$${original}\:{place}\:. \\ $$

Question Number 217958 Answers: 2 Comments: 2

a,b,c∈Z^+ and a^2 +b^2 +c^2 +ab+bc+ca=2025 find out all triplets (a,b,c).

$${a},{b},{c}\in\mathbb{Z}^{+} {and} \\ $$$${a}^{\mathrm{2}} +{b}^{\mathrm{2}} +{c}^{\mathrm{2}} +{ab}+{bc}+{ca}=\mathrm{2025} \\ $$$${find}\:{out}\:{all}\:{triplets}\:\left({a},{b},{c}\right). \\ $$

Question Number 217952 Answers: 2 Comments: 0

If x∈Z ∧y non-negative integer such that x^2 +10x+23=2^y find out x,y

$${If}\:{x}\in\mathbb{Z}\:\wedge{y}\:{non}-{negative}\:{integer} \\ $$$${such}\:{that} \\ $$$${x}^{\mathrm{2}} +\mathrm{10}{x}+\mathrm{23}=\mathrm{2}^{{y}} \\ $$$${find}\:{out}\:{x},{y} \\ $$

Question Number 217244 Answers: 1 Comments: 0

Find all two-digit numbers such that when the number is divided by the sum of its digits the quotient is 4 and the remainder is 3.

$$\:\mathrm{Find}\:\mathrm{all}\:\mathrm{two}-\mathrm{digit}\:\mathrm{numbers}\:\mathrm{such}\:\mathrm{that} \\ $$$$\mathrm{when}\:\mathrm{the}\:\mathrm{number}\:\mathrm{is}\:\mathrm{divided}\:\mathrm{by} \\ $$$$\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{its}\:\mathrm{digits}\:\mathrm{the}\:\mathrm{quotient}\: \\ $$$$\mathrm{is}\:\mathrm{4}\:\mathrm{and}\:\mathrm{the}\:\mathrm{remainder}\:\mathrm{is}\:\mathrm{3}. \\ $$

Question Number 217132 Answers: 0 Comments: 0

Find all integers n> 1 such that n divides 2^(n−1) + 3^(n−1) .

$$ \\ $$$$\mathrm{Find}\:\mathrm{all}\:\mathrm{integers}\:\:\mathrm{n}>\:\mathrm{1}\:\mathrm{such}\:\mathrm{that} \\ $$$$\:\:\mathrm{n}\:\:\mathrm{divides}\:\:\mathrm{2}^{\mathrm{n}−\mathrm{1}} \:+\:\mathrm{3}^{\mathrm{n}−\mathrm{1}} . \\ $$

Question Number 217130 Answers: 0 Comments: 0

Prove that for every integer n≥2 the number n^4 + 4^n is composite.

$$ \\ $$$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{for}\:\mathrm{every}\:\mathrm{integer}\:\:\mathrm{n}\geqslant\mathrm{2}\:\:\mathrm{the}\:\mathrm{number}\:\:\mathrm{n}^{\mathrm{4}} +\:\mathrm{4}^{{n}} \:\:\mathrm{is} \\ $$$$\mathrm{c}{o}\mathrm{mposite}. \\ $$

Question Number 217129 Answers: 1 Comments: 2

prove that if an integer n is not divisible by 2 or 3 then n^2 ≡1(mod 24)

$${prove}\:{that}\:{if}\:{an}\:{integer}\:{n}\:{is}\:{not}\:{divisible}\:{by}\:\mathrm{2}\:{or}\:\mathrm{3} \\ $$$$\:{then}\:{n}^{\mathrm{2}} \equiv\mathrm{1}\left({mod}\:\mathrm{24}\right) \\ $$

Question Number 217066 Answers: 1 Comments: 0

Find all integer x,y such that x^2 −y^2 =100

$${Find}\:{all}\:{integer}\:{x},{y}\:{such}\:{that} \\ $$$${x}^{\mathrm{2}} −{y}^{\mathrm{2}} =\mathrm{100} \\ $$

Question Number 217071 Answers: 1 Comments: 0

Find all two-digit numbers that are equal to four times the sum of their digits. Solve this using at least two different methods and verify your answers.

$$\mathrm{Find}\:\mathrm{all}\:\mathrm{two}-\mathrm{digit}\:\mathrm{numbers}\:\mathrm{that}\:\mathrm{are}\:\mathrm{equal}\:\mathrm{to}\:\mathrm{four}\:\mathrm{times}\:\mathrm{the}\:\mathrm{sum}\: \\ $$$$\mathrm{of}\:\mathrm{their}\:\mathrm{digits}.\:\mathrm{Solve}\:\mathrm{this}\:\mathrm{using}\:\mathrm{at}\:\mathrm{least}\:\mathrm{two}\:\mathrm{different}\:\mathrm{methods}\: \\ $$$$\mathrm{and}\:\mathrm{verify}\:\mathrm{your}\:\mathrm{answers}. \\ $$

Question Number 217040 Answers: 2 Comments: 0

Find all positive integers n such that n divides 2^n + 1.

$$\mathrm{Find}\:\mathrm{all}\:\mathrm{positive}\:\mathrm{integers}\:\:\mathrm{n}\:\:\mathrm{such}\:\mathrm{that} \\ $$$$\:\:\mathrm{n}\:\:\mathrm{divides}\:\:\mathrm{2}^{{n}} \:+\:\mathrm{1}.\:\: \\ $$

Question Number 217030 Answers: 1 Comments: 0

Find all positive integers n such that n + 1 divides n^2 + 1

$$\mathrm{Find}\:\mathrm{all}\:\mathrm{positive}\:\mathrm{integers}\:\:\mathrm{n}\:\:\mathrm{such}\:\mathrm{that}\:\: \\ $$$$\:\mathrm{n}\:+\:\mathrm{1}\:\:\mathrm{divides}\:\:\mathrm{n}^{\mathrm{2}} \:+\:\mathrm{1} \\ $$

Question Number 216995 Answers: 3 Comments: 0

Find all prime numbers p and q such that p^2 − q^2 = 2024

$$\mathrm{Find}\:\mathrm{all}\:\mathrm{prime}\:\mathrm{numbers}\:\:\mathrm{p}\:\mathrm{and}\:\mathrm{q}\: \\ $$$$\mathrm{such}\:\mathrm{that} \\ $$$$\mathrm{p}^{\mathrm{2}} −\:\:\mathrm{q}^{\mathrm{2}} =\:\:\mathrm{2024} \\ $$

Question Number 216912 Answers: 0 Comments: 0

Find all three-digit numbers n such that 1. n is divisible by the sum of its digits. 2. n is a perfect square.