Math moments: uses of mathematics in today’s world

# nature of mathematics

# Trigonometric Telescopic Sums and Products: A free video lecture from Mathematics Hothouse

# Questions based on Wilson’s theorem for training for RMO

1(a) Find the remainder when is divided by 17.

1(b) Find the remainder when is divided by 29.

2: Determine whether 17 is a prime by deciding if

3: Arrange the integers 2,3,4, …, 21 in pairs a and b that satisfy .

4: Show that .

5a: Prove that an integer is prime if and only if .

5b: If n is a composite integer, show that , except when .

6: Given a prime number p, establish the congruence

7: If p is prime, prove that for any integer a, and

8: Find two odd primes for which the congruence holds.

9: Using Wilson’s theorem, prove that for any odd prime p:

10a: For a prime p of the form , prove that either

or

10b: Use the part (a) to show that if is prime, then the product of all the even integers less than p is congruent modulo p to either 1 or -1.

More later,

Nalin Pithwa.

# Wilson’s theorem and related problems in Elementary Number Theory for RMO

I) Prove Wilson’s Theorem:

If p is a prime, then .

Proof:

The cases for primes 2 and 3 are clearly true.

Assume

Suppose that a is any one of the p-1 positive integers and consider the linear congruence

. Then, .

Now, apply the following theorem: the linear congruence has a solution if and only if , where . If , then it has d mutually incongruent solutions modulo n.

So, by the above theorem, the congruence here admits a unique solution modulo p; hence, there is a unique integer , with , satisfying .

Because p is prime, if and only if or . Indeed, the congruence is equivalent to . Therefore, either , in which case , or , in which case .

If we omit the numbers 1 and p-1, the effect is to group the remaining integers into pairs and , where , such that the product . When these congruences are multiplied together and the factors rearranged, we get

or rather

Now multiply by p-1 to obtain the congruence

, which was desired to be proved.

An example to clarify the proof of Wilson’s theorem:

Specifically, let us take prime . It is possible to divide the integers into pairs, each product of which is congruent to 1 modulo 13. Let us write out these congruences explicitly as shown below:

Multpilying these congruences gives the result

and as

Thus, with prime .

Further:

The converse to Wilson’s theorem is also true. If , then n must be prime. For, if n is not a prime, then n has a divisor d with is prime if and only if . Unfortunately, this test is of more theoretical than practical interest because as n increases, rapidly becomes unmanageable in size.

Let us illustrate an application of Wilson’s theorem to the study of quadratic congruences{ What we mean by quadratic congruence is a congruence of the form , with }

Theorem: The quadratic congruence , where p is an odd prime, has a solution if and only if .

Proof:

Let a be any solution of so that . Because , the outcome of applying Fermat’s Little Theorem is

The possibility that for some k does not arise. If it did, we would have

Hence, . The net result of this is that , which is clearly false. Therefore, p must be of the form .

Now, for the opposite direction. In the product

we have the congruences

Rearranging the factors produces

because there are minus signs involved. It is at this point that Wilson’s theorem can be brought to bear; for, , hence,

If we assume that p is of the form , then , leaving us with the congruence

.

The conclusion is that the integer satisfies the quadratic congruence .

Let us take a look at an actual example, say, the case , which is a prime of the form . Here, we have , and it is easy to see that and .

Thus, the assertion that is correct for .

Wilson’s theorem implies that there exists an infinitude of composite numbers of the form . On the other hand, it is an open question whether is prime for infinitely many values of n. Refer, for example:

https://math.stackexchange.com/questions/949520/are-there-infinitely-many-primes-of-the-form-n1

More later! Happy churnings of number theory!

Regards,

Nalin Pithwa

# Eight digit bank identification number and other problems of elementary number theory

Question 1:

Consider the eight-digit bank identification number , which is followed by a ninth check digit chosen to satisfy the congruence

(a) Obtain the check digits that should be appended to the two numbers 55382006 and 81372439.

(b) The bank identification number has an illegitimate fourth digit. Determine the value of the obscured digit.

Question 2:

(a) Find an integer having the remainders 1,2,5,5 when divided by 2, 3, 6, 12 respectively (Yih-hing, died 717)

(b) Find an integer having the remainders 2,3,4,5 when divided by 3,4,5,6 respectively (Bhaskara, born 1114)

(c) Find an integer having remainders 3,11,15 when divided by 10, 13, 17, respectively (Regiomontanus, 1436-1476)

Question 3:

Question 3:

Let denote the nth triangular number. For which values of n does divide

Hint: Because , it suffices to determine those n satisfying

Question 4:

Find the solutions of the system of congruences:

Question 5:

Obtain the two incongruent solutions modulo 210 of the system

Question 6:

Use Fermat’s Little Theorem to verify that 17 divides

Question 7:

(a) If , show that . Hint: From Fermat’s Little Theorem, and

(b) If , show that divides

(c) If , show that

Question 8:

Show that and . Do there exist infinitely many composite numbers n with the property that and ?

Question 9:

Prove that any integer of the form is an absolute pseudoprime if all three factors are prime; hence, is an absolute pseudoprime.

Question 10:

Prove that the quadratic congruence , where p is an odd prime, has a solution if and only if .

Note: By quadratic congruence is meant a congruence of the form with . This is the content of the above proof.

More later,

Nalin Pithwa.

# Pre RMO algebra : some tough problems

Question 1:

Find the cube root of

Question 2:

Find the square root of

Question 3:

Simplify (a):

Simplify (b):

Question 4:

Solve :

Question 5:

Solve the following simultaneous equations:

and

Question 6:

Simplify (a):

Simplify (b):

Question 7:

Find the HCF and LCM of the following algebraic expressions:

and and

Question 8:

Simplify the following using two different approaches:

Question 9:

Solve the following simultaneous equations:

Slatex x^{2}y^{2} + 192 = 28xy$ and

Question 10:

If a, b, c are in HP, then show that

Question 11:

if , prove that

Question 12:

Determine the ratio if we know that

More later,

Nalin Pithwa

Those interested in such mathematical olympiads should refer to:

https://olympiads.hbcse.tifr.res.in

(I am a tutor for such mathematical olympiads).

# Elementary Number Theory, ISBN numbers and mathematics olympiads

Question 1:

The International Standard Book Number (ISBN) used in many libraries consists of nine digits followed by a tenth check digit (somewhat like Hamming codes), which satisfies

Determine whether each of the ISBN’s below is correct.

(a) 0-07-232569-0 (USA)

(b) 91-7643-497-5 (Sweden)

(c) 1-56947-303-10 (UK)

Question 2:

When printing the ISBN , two unequal digits were transposed. Show that the check digits detected this error.

Remark: Such codes are called error correcting codes and are fundamental to wireless communications including cell phone technologies.

More later,

Nalin Pithwa.