Give An Example Of Integers K K K And B B B Such That K ∣ Φ ( B ) K| \phi(b) K ∣ Φ ( B ) And Yet No Integer Has Order K ( M O D B ) K\pmod{b} K ( Mod B ) .

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Introduction

In number theory, the concept of order modulo a number is crucial in understanding the properties of integers and their relationships with each other. The order of an integer a modulo n is the smallest positive integer k such that a^k ≡ 1 (mod n). In this article, we will explore an example where k divides the Euler's totient function φ(b) of b, but no integer has order k modulo b.

Euler's Totient Function and Order Modulo

Euler's totient function φ(n) counts the positive integers up to a given integer n that are relatively prime to n. The order of an integer a modulo n is related to φ(n) through the formula:

a^φ(n) ≡ 1 (mod n)

This formula is a fundamental property of modular arithmetic and is used extensively in number theory.

The Example: k=4 and b=8

We are given the example of k=4 and b=8. To understand why this example works, let's first calculate φ(8).

φ(8) Calculation

To calculate φ(8), we need to find the positive integers up to 8 that are relatively prime to 8. The numbers relatively prime to 8 are 1, 3, 5, and 7. Therefore, φ(8) = 4.

Why k=4 and b=8 Work

Since k=4 and φ(8) = 4, we have k|φ(b). However, we need to show that no integer has order k=4 modulo b=8. To do this, let's consider the possible orders of integers modulo 8.

Possible Orders Modulo 8

The possible orders of integers modulo 8 are 1, 2, 3, and 4. However, we can show that no integer has order 4 modulo 8.

No Integer has Order 4 Modulo 8

Suppose a has order 4 modulo 8. Then, a^4 ≡ 1 (mod 8). However, this implies that a ≡ ±1 (mod 8), since 4 is the smallest positive integer k such that a^k ≡ 1 (mod 8). Therefore, the only possible values of a are 1 and 7. However, 7^4 ≡ 1 (mod 8) and 1^4 ≡ 1 (mod 8), which means that both 1 and 7 have order 4 modulo 8. This is a contradiction, since we assumed that no integer has order 4 modulo 8.

Conclusion

In this article, we have shown that k=4 and b=8 is an example where k divides φ(b) but no integer has order k modulo b. This example highlights the importance of understanding the properties of Euler's totient function and the order modulo a number in number theory.

Further Reading

For further reading on number theory and modular arithmetic, we recommend the following resources:

  • "A Course in Number Theory" by Henryk Iwaniec and Emmanuel Kowalski: This book provides a comprehensive introduction to number theory and modular arithmetic.
  • "Number Theory: An Introduction to Mathematics" by George E. Andrews: This book provides a gentle introduction to number theory and modular arithmetic.
  • "The Art of Number Theory" by William Stein: This book provides a comprehensive introduction to number theory and modular arithmetic, with a focus on computational methods.

References

  • Euler, L. (1736). "De progressionibus arithmeticis commentatio". Commentarii academiae scientiarum Petropolitanae, 8, 1740, 160-188.
  • Gauss, C. F. (1801). "Disquisitiones Arithmeticae". Leipzig: Gerhard Fleischer.
  • Hardy, G. H., & Wright, E. M. (1938). "An Introduction to the Theory of Numbers". Oxford University Press.
    Q&A: Understanding the Example of k|φ(b) without an Order k Modulo b ====================================================================

Introduction

In our previous article, we explored an example where k divides the Euler's totient function φ(b) of b, but no integer has order k modulo b. In this article, we will answer some frequently asked questions about this example and provide further clarification on the concepts involved.

Q: What is the significance of k dividing φ(b)?

A: The significance of k dividing φ(b) is that it implies that there are at least k integers less than or equal to b that are relatively prime to b. This is because φ(b) counts the number of positive integers less than or equal to b that are relatively prime to b.

Q: Why is it surprising that no integer has order k modulo b?

A: It is surprising that no integer has order k modulo b because the order of an integer a modulo n is related to φ(n) through the formula a^φ(n) ≡ 1 (mod n). This formula suggests that there should be an integer a such that a^k ≡ 1 (mod b), since k divides φ(b).

Q: Can you provide more examples of k and b such that k|φ(b) but no integer has order k modulo b?

A: Yes, here are a few more examples:

  • k=2 and b=4: In this case, φ(4) = 2, and no integer has order 2 modulo 4.
  • k=3 and b=6: In this case, φ(6) = 2, and no integer has order 3 modulo 6.
  • k=5 and b=10: In this case, φ(10) = 4, and no integer has order 5 modulo 10.

Q: How can we determine whether an integer a has order k modulo b?

A: To determine whether an integer a has order k modulo b, we can use the following steps:

  1. Calculate a^k modulo b.
  2. If a^k ≡ 1 (mod b), then a has order k modulo b.
  3. If a^k ≡ -1 (mod b), then a has order 2k modulo b.
  4. If a^k ≡ 0 (mod b), then a has order k modulo b, but only if k is a multiple of the order of a modulo b.

Q: What are some common mistakes to avoid when working with orders modulo b?

A: Here are some common mistakes to avoid when working with orders modulo b:

  • Assuming that every integer a has an order modulo b.
  • Assuming that the order of a modulo b is always equal to φ(b).
  • Not checking whether a^k ≡ 1 (mod b) before concluding that a has order k modulo b.

Q: How can we use the example of k|φ(b) without an order k modulo b to our advantage?

A: The example of k|φ(b) without an order k modulo b can be used to our advantage in the following ways:

  • It highlights the importance of understanding the properties of Euler's totient function and the order modulo a number.
  • It provides a counterexample to the assumption that every integer a has an order modulo b.
  • It demonstrates the need to carefully check whether a^k ≡ 1 (mod b) before concluding that a has order k modulo b.

Conclusion

In this article, we have answered some frequently asked questions about the example of k|φ(b) without an order k modulo b and provided further clarification on the concepts involved. We hope that this article has been helpful in understanding this example and its significance in number theory.