Tanya Khovanova's Math Blog

The smallest positive index m such that the Fibonacci number F_m is divisible by the number p is called the rank of apparition of p. If p is prime, one can prove that any Fibonacci number that is divisible by p has an index divisible by m.

Even Fibonaccis have indices divisible by 3. That means that if for some p the rank of apparition of p is divisible by three, all the Fibonaccis that are divisible by p are even. Therefore, no odd Fibonacci divides p. I already discussed this subject in my previous post “9 Divided no Odd Fibonacci.”

Now let’s look more closely at the set of primes that divide no odd Fibonacci. The Fibonacci numbers obey the following identity: F_n+k = (1/2)(F_nL_k + F_kL_n), where L_n are Lucas numbers. From here F_3n = (1/2)F_n(L_2n + L_n²). Like Fibonacci numbers, exactly every third Lucas number is even. Hence, the parity of L_2n is the same as the parity of L_n. Hence, L_2n + L_n² is divisible by 2. Let us denote G_n = (1/2)(L_2n + L_n²).

As we have already discussed before, if no odd Fibonacci is divisible by p, p‘s rank of apparition is of the form 3n which means p divides F_3n and doesn’t divide F_n. Hence, p divides G_n. On the other hand, we can show that G_n = 5F_n² + 3(-1)ⁿ. Hence, the only common divisor that F_n and G_n can have is 3. Let us take any prime divisor s of G_n other than 3. We see that F_3n is divisible by s while F_n is not. The rank of apparition of s must be a divisor of 3n and not a divisor of n. Hence this rank is divisible by 3. Thus we can see that with the exception of 3, the set of prime divisors of elements of G_n is the set of primes that do not divide odd Fibonaccis.

Here’s a bit more info about the sequence G_n. It is sequence A047946 in the Online Encyclopedia of Integer Sequences. It is a recurrence: G_n=2*G_n-1+2*G_n-2-G_n-3. Thus, we have found a recursive sequence, elements of which have a set of prime divisors which with the exception of 3 is the set of primes that do not divide odd Fibonacci numbers.

Have you ever carried your mail from your mailbox directly to your recycle bin? If so, you feel my frustration with all this wasted paper. Each time I carry these catalogues up the stairs I think that my house should have a recycle bin next to my mailbox. Even better, maybe I can put my recycle bin at the post office, so my mailman doesn’t need to carry all this weight around. The real solution would be to call the catalog company and ask them not to send any more catalogs. For a long time the idea of being put on hold for a long time and the uncertainty of success, not to mention my usual laziness, prevented me from doing this.

You might imagine how high I jumped with joy when I heard about Catalog Choice at https://www.catalogchoice.org, the company that will request on my behalf a cessation of these mailings. Their website is nicely done and appropriately greenish. You just register and enter the catalogs you do not want. It doesn’t take much time and all the negotiation is done for you.

Up to this point, I’ve rejected 58 catalogs through this website. Only eight of them confirmed that they will honor my request. I even received a special confirmation letter from L.L.Bean. Sending a letter might be polite, but it contradicts my goal of reducing the waste of paper and of my time.

Unfortunately, there were three catalogs, including Newport News, that refused to honor my request. I was so angry that I decided to call Newport News and demand my removal from their mailing list. They agreed to my request. Can you guess what happened next? I received another catalog. I called them again. And I was removed from the list again. I received yet another catalog. I called them again. They told me that I am in their database marked as a person to whom they shouldn’t send catalogs. But the catalog they sent to me had a temporary customer number, so it is not in their system and therefore doesn’t count. According to Newport News’ logic, the check-mark in the database indicating that I am not supposed to receive catalogs supersedes the fact that I actually continue receiving them. They believe their database is more reliable than facts. What can I do? I will not buy from Newport News again.

Later, I made another interesting observation: I started receiving catalogs with my name misspelled. Do they really think that if they write Tonya instead of Tanya, I will suddenly become interested in their catalogs? We all know that the reason of the misspelling is so that they can pretend to honor my request, but still send me their catalog.

This situation is not right. I shouldn’t be getting catalogs against my will, especially after I’ve made my preferences clear. I think that we should fine companies that persist in doing this. Perhaps we can have a “Do Not Mail Me Catalogs” Registry that is similar to the government’s “Do Not Call” Registry.

Meanwhile, if you are like me and care about the environment, you too can sign up at the Catalog Choice.

I was very proud of myself when I started receiving tons of comments for my math blog. Many of the comments were quite flattering. For example:

You are getting better and better. Congrats, dude.

I know that in English you can sometimes replace “ladies” with “guys”, but I wasn’t sure if being called a dude constitutes a compliment.

I got suspicious, however, when I received this:

Get real! It is interesting, but you never give proofs.

This comment was placed on my Fibonacci entry, where I think I gave more than enough proofs. So I read all the small print that accompanied the comment. It appears that “sexy-girls-in-chains” were extremely excited about the divisibility of Fibonacci numbers.

I am used to my everyday winnings of Millions of Dollars in the UK lottery, but this was something new and different. They caught me off-guard. Here I was, so proud of the public’s positive reactions, only to realize that it was some automated program. Some computer sending spam caused me to have strong emotions. I was upset that I was caught. Sigh.

What can I do? Just laugh. We can laugh together at “girls-deflowered” who were interested in Quantifying Favors, at “is-your-penis-small” who commented on Losing the Lottery, at “squirting-vibrating-realistic-anal-dildo” who was impressed with my Fantasy Future and at “teenage-girls” who associated themselves with Numbers Needing Sponsors.

One of the latest comments was:

We can professionally say that this information is objective and true and of highest quality.

It was signed by “pissing-ladies.” Am I proud or what?

Here are recent additions to my math jokes collection:

* * *

During a lecture to his students, a military instructor says, “There is a 40% chance that we will hit our target.”
One student asks, “What happens if we aim away from the target?”
The instructor replies, “Logically, we would have a 60% chance of hitting the target.”

* * *

“Do you know that 67% of people are not capable of doing simple arithmetic?”
“I belong to the other 23%.”

* * *

What is so special about 6.9?
It is 69 ruined by a period.

* * * (submitted by Irene Ogievetskaya)

Teacher: Solve the equation: x + x + x = 9.
Student: x = 3, 3, and 3.

* * *

Teacher: What is 2k + k?
Student: 3000!

I stumbled upon an article in the Boston Globe (February 29, 2008) titled “Study: Spanking children affects their sex lives as adults.” Here are some quotes:

New research by a University of New Hampshire domestic abuse expert says spanking children affects their sex lives as adults. …[C]hildren who are spanked are more likely as adults to coerce partners to have sex, to have unprotected sex and to have masochistic sex.

The classical method to prove scientifically that spanking affects something is to find many parents of newborn identical twins and persuade them to treat their children the same way, with the exception that they spank one and do not spank the other. The researchers should then compare these twins in their adulthood. Such a project is impossible, as parents are likely to start feeling guilty towards the child they spank, for you can’t separate spanking from the whole package of how parents treat their children.

I decided to study the study. I found a more detailed description of the study in the Concord Monitor. As I suspected, the study was a survey. The survey found a correlation between spanking and “undesirable” sexual behavior. As every statistician knows, correlation doesn’t prove causality.

Here’s another quote from the study: “The best-kept secret in child psychology is that children who were never spanked are among the best behaved.” Did it occur to anyone that the best behaved children do not need spanking?

Could it be that parents who spank their children are tired, impatient and less loving? Could it be that not being loved as a child affects your sexual behaviour as an adult much more than spanking?

Could it be that parents who spank their children are more aggressive in general? Could it be that they pass their aggressive genes to kids and their kids’ aggressive behavior is related not only to upbringing, but to genetics?

Do not get me wrong. I do think that spanking is bad. I am saying that the study doesn’t prove that spanking is affecting anyone’s future sex life.

I am surprised that so many magazines republished the article without thinking. Now all the country is fooled into believing that the easy way to improve their kids’ future sex life is to stop spanking.

Go ahead! Stop spanking. Love your children too.

I tried to enroll on a website recently, but they didn’t allow me to continue without choosing five security questions out of about ten samples they supplied. I started in good faith to do what they asked.

Question: What is your father’s middle name?
Answer: They do not have middle names in Russia; they have something called “otchestvo” and I know seven different ways to spell my father’s.

Question: What is the name of the street on which you were born?
Answer: I am glad it was not Lenin Street, but it was equally bad. Besides, it was renamed and I am not sure which name to choose.

Question: What is the name of your high school?
Answer: Finally, an easy question. In Russia we didn’t have names, but rather numbers for schools. I happily entered 444, and oops — the applet wouldn’t accept numbers.

I couldn’t find five questions that I could answer uniquely and reliably. I felt that the designers of these questions were clueless and disrespectful to other cultures. Then I thought about whether I really wanted some creepy database to know the name of my best friend. No, I didn’t.

Now I have established a file where I put the answers to security questions and I can have all the fun I want with my new biography. I can name my first dog Tom Cruise and have my wedding date be 20 years before I was born. I can name my husband Freedom Of Speech and my city of birth IHateSecurityQuestions. Maybe next time I will switch: Freedom Of Speech will be my dog and Tom Cruise my husband.

If you are lazy like me, you can choose your questions so you have the same answer for everything. This way you do not need to type much into your file. For example, you can name your city, your cat and your best friend George Washington. Or, if you are really lazy, God.

I just read the following in Women’s Health Magazine (March, 2008; page 54): Visa conducted a study of 100,000 fast food restaurant transactions. They found that people who pay with credit cards spend 30% more on food than people who pay cash.

The article concludes with the suggestion to pay cash, so you spend less and lose weight.

My question is: Who is more incompetent, Visa or Women’s Health Magazine?

Perhaps people who do not have credit cards are poorer and more price-conscious; hence, they spend less on food. This might explain the correlation. Here’s another possible explanation: people who are ordering for large groups might prefer to pay with a credit card. Or, maybe stores do not like using credit cards for small transactions, so they encourage people to pay cash for modest orders.

The main rule of statistics is that correlation doesn’t mean causality.

There are several possible answers to my question about incompetence:

• The study wasn’t described correctly in the WH magazine. In this case we can’t say much about the competence of Visa, but WH looks bad.
• The study was described correctly, but the conclusion belongs to WH. In this case Visa is innocent and WH is incompetent.
• The study was described correctly and Visa suggested the conclusions. In this case both are incompetent  — Visa for its conclusions and WH for printing them.

It could well be that paying cash makes you stingier, or at least more price-conscious, but I can’t trust Women’s Health Magazine any more. One thing I know for sure is that math can help you lose weight. Math allows you to differentiate a good study from a dumb study.

Recently I was buying gas at a gas station. I was paying with my credit card and the machine asked for my zip code. If you read my previous post, you know what happened: I got furious. First, I tried 00000 as a zip code. The machine was smart enough to tell me that it was invalid. No matter what combination I tried, it wouldn’t accept another zip code.

Finally, the machine got frustrated with me and printed on its screen that I needed to talk to the cashier. Then, I argued with the cashier. He could have suggested that I pay cash, but he didn’t. I left without gas.

Luckily, for me I had dinner that day with my son, Alexey Radul, and his friend, Grem. Grem explained to me that gas stations are not collecting customers’ zip codes. They use zip codes as a security measure for checking that the credit card is not stolen. I guess that means I behaved exactly like a person who stole a card. Protecting your own privacy sometimes makes you look like a thief.

After dinner, I went back to the same gas station to conduct some experiments. This time I was armed with several valid zip codes. It didn’t work. Grem was right — only the zip code corresponding to the card could have worked. I looked like a thief again. I paid for my gas with cash and left.

The next day I called my credit card company and asked them about this. The manager I talked to told me the same thing as the manager Grem talked to. Apparently credit cards do not give your zip code information to gas stations. Your zip code is used instead of a pin number to verify that your credit card is not stolen. So in this case you do not need to worry that you are providing extra information, since credit card companies know your zip code anyway.

Nowadays, supporting a database is cheap. Computer storage is cheap, too. This gives companies an opportunity to collect more and more information about us. If you think, as I do, that this is an invasion of your privacy, here are some ways to resist.

When you buy something over the Internet or through a catalog, they ask for both your email address and telephone number. They may need a way to contact you in case something happens with your order, but they do not need both. When you are ordering online and their default contact is by email, they do not need your phone number. If the website requires your phone number, you can put in a fake number. Of course, you are a nice person and do not want to provide some innocent soul’s phone number instead of yours. Here is the perfect solution. Put the number 555-5555 as your home number with any area code.

The phone numbers of the format 555-xxxx are reserved for the movie industry. That is, if Hugh Grant calls Julia Roberts in a movie, there would be hundreds of bored or not very smart people who would try to call the same number Hugh dialed hoping to talk to Julia Roberts. For these situations, the movie industry reserves all the numbers of the form 555-xxxx. This way they guarantee that all of these fans will not bother a real person. So you can use these numbers without any guilt.

If you are ordering by phone, they might see your number on the caller ID. In this case, you can always say that you do not have an email address. You can also use a one-time email address offered through Sneakmail or AddressGuard at Yahoo.

Store shopping cards also scare me very much. When you use your store shopping card, they know exactly what and when and in what amounts you are buying. If you do not want anyone to know that you are buying 100 Tylenol pills a month, do not use your store card, and consider paying cash.

My friend Sam Steingold suggested I try card swapping. You have a CVS card and your friend has a CVS card — you can swap them. CVS’s database will register that you quit buying Tylenol in Boston, but started buying cigarettes in Atlanta. If you continue swapping, CVS’s database will be totally confused. The good part of this idea is that if someone tries to hold your purchases against you, you have a way to prove that you are not responsible.

The disadvantage of card swapping, is that for the transition time you lose targeted coupons. Your friend in Atlanta will get all the Tylenol coupons he doesn’t need. But you still will be able to buy sale items with discounts.

Here’s what I did – I put another last name on my CVS card. They didn’t notice. If they were to notice, I would have told them that I am in process of changing my last name to my newly acquired husband’s last name and would ask for newlyweds’ coupons.

Sometimes when you buy things, they ask you for your phone number at the cash register. It is even worse than shopping cards. They have your information on file without giving you your discounts. Just remember: you can always refuse. Or if you’re not comfortable refusing, let us all agree to give the same number: (area code)-555-5555. Let their analysts wonder why the same person is buying morning-sickness pills in one store and condoms in another.

My Number Gossip page is on hold. I am out of work and can only afford to spend time on things that can bring me my next job. As a result, numbers suffer. If you would like to support the website, consider donating to Number Gossip. With your donations, I will be able to spend more time on the website. There are many things I would like to do. Here are the three most important and fun areas that I would chose to work on.

1. Do you know that 40(forty) is the only number whose constituent letters appear in alphabetical order? I have about 1,000 unique number properties like this on my number gossip website. I also have about 600 properties in my database that need to be checked before adding them to the website. Being a perfectionist I double-check and recheck all the properties. As a result, my webpage is quite rigorous and reliable. As a downside I need a lot of time to add new unique properties to the website and 600 of such properties are waiting impatiently to join their comrades in the public view.
2. Have you heard about aspiring or untouchable numbers? What about practical or perfect numbers? There 50 famous properties like that that my website currently allows you to check for your favorite number. For example, 33 is deficient, evil, lucky and odd at the same time. There are many other properties I plan to add in the future. If you would like to know whether or not your favorite number is brilliant, fortunate or primeval, consider donating to Number Gossip.
3. Currently the limit for numbers you can input is 10,000. Some of the basic properties are very difficult to check. The expansion is not trivial and will require significant tuning of my algorithms. Even so, I would like to do that.

I’ve devoted many years to this project, and now I need some financial help. If you know a person or a company who wants to sponsor numbers, please ask them to contact me at (tanyakh at yahoo).