Understanding Prime Factors of ( p^4 - 1 )

When it comes to the expression ( p^4 - 1 ), the total number of its prime factors is not fixed. This article explores the complexity and variability of prime factors in such expressions, including the role of the Omega function and specific prime values of ( p ) .

General Analysis of ( p^4 - 1 )

For any odd prime ( p ), the expression ( p^4 - 1 ) can be factored as follows:

p^4 - 1  (p^2   1)(p   1)(p - 1)

This factorization reveals that ( p^4 - 1 ) has at least three distinct prime factors, as each term can potentially introduce a new prime factor. However, the exact number of prime factors can vary depending on the specific value of ( p ).

Role of the Omega Function

The Omega function, denoted as ( Omega(n) ), counts the number of distinct prime factors of an integer ( n ) (counting each prime factor according to its multiplicity).

In the context of ( p^4 - 1 ), the Omega function can provide insights into the number of distinct prime factors of ( n ). Let's analyze some specific cases:

Prime Numbers of ( p )

Consider the case when ( p ) is a prime number. For smaller values of ( p ), we can use a brute force approach to determine the number of distinct prime factors:

00:08 gp for p  100 printp, Omega(p)2 23 24 35 36 37 38 49 310 311 412 413 514 415 316 417 418 419 420 421 522 523 524 325 426 327 528 429 530 431 532 533 434 635 436 437 538 539 440 441 542 443 644 545 446 547 648 449 450 551 452 553 554 455 556 557 558 559 560 461 562 563 464 665 466 467 668 569 570 571 572 573 674 475 576 677 678 579 580 481 582 483 784 485 586 687 588 589 690 491 692 693 594 595 496 597 598 699 5100 5

From the above results, we can observe that for different values of ( p ), the number of distinct prime factors varies. This demonstrates the variability of prime factors in ( p^4 - 1 ).

Specific Analysis of ( p^4 - 1 )

Now, let's delve deeper into the specific expression ( p^4 - 1 ) and analyze its prime factors:

For ( p 2 ):

( p^4 - 1 2^4 - 1 16 - 1 15 3 times 5 ). Here, there are exactly two distinct prime factors: 3 and 5.

For other prime values of ( p ):

The expression ( p^4 - 1 ) can be further factored into ( p^2 1 ), ( p 1 ), and ( p - 1 ). Each of these terms can introduce distinct prime factors, and the exact number depends on the value of ( p ).

For example:

( p 3 ): ( p^4 - 1 3^4 - 1 81 - 1 80 5 times 2^4 ). Here, the distinct prime factors are 2 and 5.( p 5 ): ( p^4 - 1 5^4 - 1 625 - 1 624 2^4 times 3 times 13 ). Here, the distinct prime factors are 2, 3, and 13.

From these examples, we can see that the number of distinct prime factors varies significantly depending on the specific value of ( p ).

Potential Prime Factors

More specifically, the expression ( p^4 - 1 ) can have additional prime factors beyond the three base terms. For instance:

( p^4 - 1 ) can be written as ( (p^2 1)(p 1)(p - 1) ). Each term can introduce additional prime factors, and the exact number depends on the properties of ( p ).

The term ( p^2 1 ) can introduce new prime factors, and so can ( p 1 ) and ( p - 1 ).

For example, consider ( p 11 ):

11^4 - 1  14641 - 1  14640  2^4 times 3 times 5 times 61

In this case, the prime factors are 2, 3, 5, and 61.

Conclusion

The number of prime factors of ( p^4 - 1 ) is not fixed and varies depending on the value of ( p ). For ( p 2 ), the expression simplifies to ( 15 3 times 5 ), having two distinct prime factors. For other prime values of ( p ), the number of prime factors can range from a minimum of three to a higher number depending on the specific ( p ). The Omega function can help in determining the exact number of distinct prime factors.