The document discusses several unsolved problems in mathematics, including Catalan's Conjecture, Twin Primes Conjecture, Goldbach's Conjecture, the existence of odd perfect numbers, and the Collatz Conjecture. It outlines questions posed by each conjecture and provides examples to illustrate these mathematical concepts. The author, K. Thiyagu, is an assistant professor at the Department of Education, Central University of Kerala.

1. Unsolved Mathematics Problems
K.THIYAGU
Assistant Professor
Department of Education
Central University of Kerala, Kasaragod

2. Catalan’s Conjecture
An integer is a perfect power if it is of the form m^n where m and n are
integers and n>1. It is conjectured that 8=2^3 and 9=3^2 are the only
consecutive integers that are perfect powers. The conjecture was finally proved
by Preda Mihailescu in a manuscript privately circulated on April 18, 2002.
Are 8 and 9 the only consecutive perfect powers?
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3. Twin Primes Conjecture
A prime number is an integer larger than 1 that has no divisors other than
1 and itself. Twin primes are two prime numbers that differ by 2.
For example, 17 and 19 are twin primes.
Are there an infinite number of twin primes?
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4. Goldbach’s Conjecture
Is every even integer larger than
2 the sum of two primes? A prime number is an
integer larger than 1 whose
only positive divisors are 1
and itself. For example, the
even integer 50 is the sum of
the two primes 3 and 47.
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5. Odd Perfect Numbers
A perfect number is a positive
integer that is equal to the sum of
all its positive divisors, other than
itself. For example, 28 is perfect
because 28=1+2+4+7+14.
Are there any odd perfect numbers?
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6. The Collatz Conjecture
Start with any positive integer.
Halve it if it is even; triple it and
add 1 if it is odd. If you keep
repeating this procedure, must you
eventually reach the number 1?
For example, starting with the
number 6, we get: 6, 3, 10, 5, 16, 8,
4, 2, 1.
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7. Expressing 3 as the Sum of Three Cubes
The number 3 can be written as
1^3+1^3+1^3
and also as
4^3+4^3+(-5)^3.
Is there any other way of
expressing 3 as the sum of three
(positive or negative) cubes?
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8. Diophantine Equation of Degree Five
Are there distinct positive integers,
a, b, c, and, d such that
a^5+b^5=c^5+d^5?
It is known that
1^3+12^3=9^3+10^3 and
133^4+134^4=59^4+158^4,
but no similar relation is known for fifth powers.
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9. Thank You
K.THIYAGU
Assistant Professor
Department of Education
Central University of Kerala, Kasaragod
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