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Number TheoryQuestion and Answers: Page 2

Question Number 216538 Answers: 0 Comments: 4

Find all integer solutions of 3^m =2n^2 +1. I only found m=1, 2, 5 by computer from m=1 to m=30000. Is there any greater solutions?

$$\mathrm{Find}\:\mathrm{all}\:\mathrm{integer}\:\mathrm{solutions}\:\mathrm{of} \\ $$$$\mathrm{3}^{{m}} =\mathrm{2}{n}^{\mathrm{2}} +\mathrm{1}. \\ $$$$ \\ $$$${I}\:{only}\:{found}\:{m}=\mathrm{1},\:\mathrm{2},\:\mathrm{5}\:{by}\:{computer} \\ $$$${from}\:{m}=\mathrm{1}\:{to}\:{m}=\mathrm{30000}. \\ $$$${Is}\:{there}\:{any}\:{greater}\:{solutions}? \\ $$

Question Number 215272 Answers: 1 Comments: 0

{ ((abac^(−) =(dc^(−) )^2 )),((d=((ab^(−) )/c))),((c^2 =ac^(−) )) :} abac^(−) =?

$$\begin{cases}{\overline {{abac}}=\left(\overline {{dc}}\right)^{\mathrm{2}} }\\{{d}=\frac{\overline {{ab}}}{{c}}}\\{{c}^{\mathrm{2}} =\overline {{ac}}}\end{cases} \\ $$$$\overline {{abac}}=? \\ $$

Question Number 214638 Answers: 1 Comments: 0

if the sum of three prime numbers is 130, what is the possible maximum of their product?

$${if}\:{the}\:{sum}\:{of}\:{three}\:{prime}\:{numbers} \\ $$$${is}\:\mathrm{130},\:{what}\:{is}\:{the}\:{possible}\: \\ $$$${maximum}\:{of}\:{their}\:{product}? \\ $$

Question Number 213756 Answers: 1 Comments: 0

Question Number 213663 Answers: 1 Comments: 0

Given a,b,c is natural numbers such that (a−b)(b−c)(c−a)=a+b+c. find min value of a+b+c

$$\:\:\mathrm{Given}\:\mathrm{a},\mathrm{b},\mathrm{c}\:\mathrm{is}\:\mathrm{natural}\:\mathrm{numbers} \\ $$$$\:\:\mathrm{such}\:\mathrm{that}\:\left(\mathrm{a}−\mathrm{b}\right)\left(\mathrm{b}−\mathrm{c}\right)\left(\mathrm{c}−\mathrm{a}\right)=\mathrm{a}+\mathrm{b}+\mathrm{c}. \\ $$$$\:\:\mathrm{find}\:\mathrm{min}\:\mathrm{value}\:\mathrm{of}\:\mathrm{a}+\mathrm{b}+\mathrm{c}\: \\ $$

Question Number 212999 Answers: 1 Comments: 0

Find the number of non zero integer solution (x,y) to the equation ((15)/(x^2 y)) + (3/(xy)) − (2/x) = 2

$$\:\:\:\mathrm{Find}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{non}\:\mathrm{zero}\:\mathrm{integer}\: \\ $$$$\:\mathrm{solution}\:\left(\mathrm{x},\mathrm{y}\right)\:\mathrm{to}\:\mathrm{the}\:\mathrm{equation}\: \\ $$$$\:\:\:\:\frac{\mathrm{15}}{\mathrm{x}^{\mathrm{2}} \mathrm{y}}\:+\:\frac{\mathrm{3}}{\mathrm{xy}}\:−\:\frac{\mathrm{2}}{\mathrm{x}}\:=\:\mathrm{2}\: \\ $$

Question Number 212242 Answers: 0 Comments: 7

Help

$$\mathrm{Help} \\ $$

Question Number 212201 Answers: 1 Comments: 0

find the last two digits of 9^9^9 ?

$$\:\:\:\:\:\:\:\mathrm{find}\:\mathrm{the}\:\mathrm{last}\:\mathrm{two}\:\mathrm{digits}\:\mathrm{of}\:\mathrm{9}^{\mathrm{9}^{\mathrm{9}} } \:? \\ $$

Question Number 211943 Answers: 2 Comments: 1

2^(m−1) =1+mn m, n ∈Z

$$\mathrm{2}^{{m}−\mathrm{1}} =\mathrm{1}+{mn} \\ $$$${m},\:{n}\:\in\mathbb{Z} \\ $$

Question Number 211635 Answers: 1 Comments: 0

Question Number 211595 Answers: 1 Comments: 0

Question Number 211456 Answers: 2 Comments: 0

Question Number 211455 Answers: 2 Comments: 0

Question Number 211454 Answers: 2 Comments: 0

Question Number 211453 Answers: 1 Comments: 0

Question Number 211445 Answers: 1 Comments: 0

Question Number 211443 Answers: 1 Comments: 0

Question Number 208506 Answers: 2 Comments: 0

What is the smallest number which must added to 9454351626 so that it will become divisible by 11?

Question Number 207713 Answers: 2 Comments: 0

Question Number 207436 Answers: 1 Comments: 2

a, b∈N_+ , ((b+1)/a)+((a+1)/b)∈Z. Prove that (a, b)≤(√(a+b.))

$${a},\:{b}\in\mathbb{N}_{+} ,\:\frac{{b}+\mathrm{1}}{{a}}+\frac{{a}+\mathrm{1}}{{b}}\in\mathbb{Z}.\:\mathrm{Prove}\:\mathrm{that}\: \\ $$$$\left({a},\:{b}\right)\leqslant\sqrt{{a}+{b}.} \\ $$

Question Number 207423 Answers: 2 Comments: 0

⇃

$$\:\:\:\:\downharpoonleft\underline{\:} \\ $$

Question Number 207060 Answers: 2 Comments: 0

⇃

$$\:\:\downharpoonleft\underline{\:} \\ $$

Question Number 207021 Answers: 1 Comments: 1

If a_(n+1) =2−5a_n and a_4 = −8 prove that a_(43) −a_(30) divisible by 5

$$\:\:{If}\:\:{a}_{{n}+\mathrm{1}} =\mathrm{2}−\mathrm{5}{a}_{{n}} \:{and}\:{a}_{\mathrm{4}} =\:−\mathrm{8} \\ $$$$\:\:{prove}\:{that}\:{a}_{\mathrm{43}} −{a}_{\mathrm{30}} \:{divisible}\:{by}\:\mathrm{5} \\ $$

Question Number 206999 Answers: 3 Comments: 1

Prove that 6^(20) −1 = 0 (mod 7)

$$\:\:\:\mathrm{Prove}\:\mathrm{that}\:\mathrm{6}^{\mathrm{20}} −\mathrm{1}\:=\:\mathrm{0}\:\left(\mathrm{mod}\:\mathrm{7}\right) \\ $$

Question Number 206912 Answers: 3 Comments: 0

Question Number 205922 Answers: 2 Comments: 0

x = 2 (mod 7) x=3 (mod 4) x=?

$$\:\:\:{x}\:=\:\mathrm{2}\:\left({mod}\:\mathrm{7}\right) \\ $$$$\:\:\:{x}=\mathrm{3}\:\left({mod}\:\mathrm{4}\right) \\ $$$$\:\:\:{x}=? \\ $$

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