Daily Test - Page 21

LoneRanger · #**301** 04-19-2008, 12:44 PM

19 April 2008:

THE SOLUTION:

Special class of numbers. For many years, mathematicians have studied this cool class of numbers. Here’s how to understand them. If a number is less than the sum of its proper divisors, it is called abundant. (A positive proper divisor is a positive divisor of a number n, excluding n itself.) As an example, the proper divisors of 12 are 1, 2, 3, 4, and 6. And these proper divisors add up to 16. The number 12 is less than 16, so 12 is abundant. The first few abundant numbers are 12, 18, 20, 24, 30, 36, . . . The first odd abundant number is 945. (Its prime factorization is 945 = 3 x3 x 3 × 5 × 7, and the sum of its factors is 975.)

Exclusionary squares. I have a particular penchant for an unusual class of numbers called “exclusionary squares,” such as the very special number 639,172. It turns out that this is the largest six (6) integer with distinct digits whose square is made up of digits not included in itself: (639,172)(639,172) = 408,540,845,584. Can you find the only other six-digit example?

Do any exclusionary cubes or exclusionary numbers of higher orders exist?

ιи¢αяиαтισи · #**302** 04-19-2008, 11:06 PM

(203,879)(203,879) = 41,566,646,641

LoneRanger · #**303** 04-20-2008, 02:37 PM

20 April 2008:

THE SOLUTION:

Exclusionary squares. The other 6-digit example is (203,879)(203,879) = 41,566,646,641. Problems like this seem to be best solved using brute-force computer methods. If we do not require the digits to be distinct,

Ilan Mayer has found several exclusionary cubes, such as (6,378) raised to the power of 3 = 259,449,922,152 and (7,658) raised to the power of 3 = 449,103,134,312. If we do not require that all the digits be unique, then we can find exclusionary squares of any length; for example, we can experiment with strings of 3s, such as (3,333,333) raised to the power of 2 = 11,111,108,888,889. German Gonzales he has found 168,569 exclusionary numbers from 1 to 1,000,000 of various orders.

For example, here is an exclusionary number of the 83rd order: (2) raised to the power of 83 = 9, 671, 406, 556, 917, 033, 397, 649, 408.

The grand search for isoprimes. Note that 11 is an isoprime, a prime number with all digits the same. (A prime number is divisible only by
itself and 1.) Do any other isoprimes exist in base 10?

Similarly, 101 is an oscillating bit prime (base 10). Do any others exist?
For example, 10,101 is not prime. Neither is 1,010,101.

LoneRanger · #**304** 04-21-2008, 01:04 PM

21 April 2008:

THE SOLUTION:

The grand search for isoprimes. Here is a list of other isoprimes (in base 10): 11; 111; 1,111,111,111,111,111,111; and 11,111, 111,111,111,111,111,111. In the world of factoring and primality testing, 11 is also called a repunit (repeated unit) prime. All repunit primes in base 10 can only be composed of 1's. The next such number has 317 digits; The next such number has 1,031 digits. After that, the next two isoprimes that are believed to be prime, but are not proven such, contain
49,081 digits and 86,453 digits. Chris Caldwell has interesting Web sites on prime numbers for further exploring: primes.utm.edu/ and primes.utm.edu/glossary/page.php/Repunit.html.

Regarding the oscillating bit prime, in 1991, Harvey Dubner discovered
a prime number with a total of 5,114 digits that is composed of only 1s and zeros. The precise number is (10{raised to the power of 5114} - 10{raised to the power of 2612} + 9)/9.

Amazing. I do not know whether the 0s and 1s oscillate in any particular pattern in this large number.

Triangle of the Gods. An angel descends to Earth and shows you the following simple progression of numbers:

The angel will let you enter the Heavenly Abode if you can determine what is the smallest prime number of this kind. Can you do so?

LoneRanger · #**305** 04-22-2008, 12:27 PM

22 April 2008:

THE SOLUTION:

Triangle of the Gods. By computer search, one can find the following smallest prime number of this kind in Row 171:

123456789012345678901234567890
123456789012345678901234567890
123456789012345678901234567890
123456789012345678901234567890
123456789012345678901234567890
123456789012345678901

The largest known prime number of this kind occurs in Row 567 and ends in the digit 7.

When you perform such searches, note that you can immediately eliminate numbers ending in the even digits and the number 5.

We can ask many questions. What percentage of prime numbers do you expect as we scan more rows in the mysterious triangle? If you could add one digit to the beginning of each number in order to increase the number of primes, what would it be? If you could add one digit to the end of each number in order to increase the number of primes, what would it be?

Body weights. What would happen if everyone’s body weight was quantized and came in multiples of pi. pounds?

LoneRanger · #**306** 04-23-2008, 12:12 PM

23 April 2008:

THE SOLUTION:

Body weights. This means that if you gained or lost weight, you would not change weight smoothly, but your weight would jump up or down by increments of 3.1415 . . .pounds. The largest biological effect of this
strange quantization would be for the newborn, where a 3-pound difference would have the most profound and perhaps fatal effect. In
other words, if this quantization became commonplace, many newborns would die. Could a premature infant weighing π pounds survive?

(Of course, I’m not implying that there is something special about pi in this question, because a 3-pound quantization would have similar effects.)

Jesus and negative numbers. Would Jesus of Nazareth or any person living in His era ever have worked with a negative number, like -3?

LoneRanger · #**307** 04-24-2008, 01:13 PM

24 April 2008:

THE SOLUTION:

No. The concept of negative numbers started in the seventh century. At this time, we first see negative numbers used in bookkeeping in India. The earliest documented evidence of the European use of negative numbers occurs in the Ars magna, published by the Italian mathematician Girolamo Cardano in 1545.

Al-Khwarizmi, who was born in Baghdad, discovered the rules for algebra around A.D. 800. Obviously, there is quite a bit of surprisingly simple mathematics that was not around in Jesus’s time.

What kinds of written numbers did Jesus of Nazareth, or a comparable
figure of his era, use? Did these people use numbers that looked like the numbers we use today?

LoneRanger · #**308** 04-25-2008, 02:32 PM

25 April 2008:

THE SOLUTION:
Some scholars, have claimed that Jesus spoke Aramaic, and we expect that Jesus used the Aramaic/Hebrew number system, where alphabetic characters also served as their numbers. Because some of the apocryphal and the pseudepigraphic infancy gospels tell tales of Jesus having discussed the symbolism of the Greek and related alphabets, one might also argue that he could have written using the Greek number system, which likewise used its alphabet for numerical digits.

If one considers the text of the New Testament as definitive, reliable, or historical, all numbers that appear in passages with references to Jesus in the four gospels are written out in Greek (e.g., eis/mian [one], duo/duos [two], treis/trisin [three], tessares [four], hex [six], hepta [seven], okto [eight], heptakis [seven times], ennea [nine], deka [ten], eikosi pente [twenty-five], triakonta [thirty], hekaton [one hundred], hebdomekontakis heptai [seventy times seven], dischilioi [two thousand], pentakischilioi [five thousand], etc.). Most numbers in the text of the Bible tend to be written out, though there are a few exceptions, such as the infamous 666 of the Apocalypsis, written with the three Greek letters chi, xi, and the antiquated sigma. In the Greek numeral system, the letter chi has a value of 600, xi 60, and the sigma/digamma a value of 6, so that the three letters appearing together as a number have the combined value of 666.

Jesus and multiplication. Could Jesus of Nazareth multiply two numbers?

LoneRanger · #**309** 04-26-2008, 11:57 AM

26 April 2008:

THE SOLUTION:

Jesus and multiplication. Very likely. In Matthew 18:22, we find, “legei auto ho Iesous Ou lego soi eos heptakis all’eos hebdomekontakis epta.” Or, in Jerome’s Vulgate, “dicit illi Iesus non dico tibi usque septies sed usque septuagies septies.” Today we translated this as “Said Jesus: To you I say not ‘til seven times,’ but ‘until seventy times seven.’”

Because both seven and seventy can have symbolic meanings, the meaning may not be literal, but, nevertheless, it is an example of multiplication.

The Bible does not make it clear whether Jesus or his listeners would have been able to give the exact answer. Much earlier, in Leviticus 25:8, we find “Seven weeks of years shall you count—seven times seven years—so that the seven cycles amount to forty-nine years.” Therefore, we know these people could do at least 7 × 7. However, we must not lose sight of the possibility that the biblical translators introduced the terms.

In addition, conversion between monetary systems like Roman sesterces, Jewish shekels, and Persian darii probably required notions of
multiplication and division. Jesus was probably aware of the concept of debts and interest charged on debts.

Jesus would not have used a symbol for zero, because neither the Hebrew, the Aramaic, nor the Greek number systems had a character representing the number 0, as it was not required by their non positional number systems.

The digits of pi. Is it true that I can find consecutive digits, like 1, 2, 3, . . .1,000,000, all neatly in a row in the decimal digits of pi?

LoneRanger · #**310** 04-27-2008, 01:00 PM

27 April 2008:

THE SOLUTION:

2.24 Certainly, if we assume that modern mathematical conjectures are
correct. Pi contains an endless number of digits with what mathematicians conjecture to be a “normal” or “patternless” distribution. We can even search for some of the first few consecutive runs, using computer searches that are available on the Web. The string 123 is found at position 1924, counting from the first digit after the decimal point. The “3.” is not counted. The string 1234 is found at position 13,807; 12345 is found at position 49,702; and so forth. You can do further searches of this kind at Dave Anderson’s π Web site:
The Pi-Search Page.

Adding numbers. It would be a tough job to add all the numbers between 1 and 1,000. What formula would you use to do this quickly?

LoneRanger · #**311** 04-28-2008, 02:38 PM

28 April 2008:

THE SOLUTION:

The mathematical prodigy Karl Friedrich Gauss (1777-1855), the son of a bricklayer, discovered that he could sum the numbers from 1 to n using
the formula n(n + 1)/2. Thus, if we want to sum 1 to 1,000, we simply compute 1,000 X (1,001)/2 = 500,500.

Little Gauss demonstrated his approach at age ten, when he quickly solved a problem that had been assigned by a teacher to keep the class busy. The teacher had asked the students to find the sum of the first 100 integers, and he was amazed that Gauss could add the terms so quickly. In fact, the teacher assumed that Gauss was wrong.

The mystery of 0.33333. We all know that 1⁄3 = 0.3333. . . repeating. Multiplying both sides of the equation by 3, we find that 1 = 0.9999 . . .
How can this be?

LoneRanger · #**312** 04-29-2008, 12:22 PM

29 April 2008:

THE SOLUTION:

The reason that we find 1 = 0.9999 . . . is that it is true. There are numerous mathematical ways to show this, that involve the sum of an infinite series, but my favorite way doesn’t require too much math.

Consider that any two distinct (different) real numbers must have another number in-between them. However, there is no number between 1 and 0.9999 . . . Thus, 1 and 0.9999 . . . are not different numbers.

Mystery sequence. What is the missing number in the following sequence? No numbers may repeat in this sequence.

13, 24, 33, 40, 45, 48, ?

LoneRanger · #**313** 04-30-2008, 01:46 PM

30 April 2008:

THE SOLUTION:

The missing number is 49. To create this sequence, I listed the numbers 1 through 13. Underneath this list, I
listed the numbers 13 through 1. Then, I just multiplied the numbers in each column:

1 2 3 4 5 6 7 8 9 10 11 12 13
13 12 11 10 9 8 7 6 5 4 3 2 1
13 24 33 40 45 48 49. . . . . . . . . . . . . . . . . . .

You can also solve this another way, simply by adding 11, 9, 7, and so forth. These numbers represent the differences between consecutive terms.

Strange code. If ..--- + ... .- equals -... ., what does .---- + ..--- equal?

LoneRanger · #**314** 05-01-2008, 02:24 PM

01 May 2008:

THE SOLUTION:

Strange code.: .--, obviously. Here, we are using the Morse code, invented by Samuel
Morse (1791-1872), in which letters and numbers are represented by dots and dashes:
(0, -----), (1, .----), (2, ..---), (3, ...--),
(4, ....-), (5,.....), (6, -....), (7, --...), (8, ---..),
and (9, ----.).

What number comes next?
1, 9, 17, 3, 11, 19, 5, 13, 21, 7, 15, ?

Acc3ss · #**315** 05-01-2008, 03:29 PM

23 ... m i right ??