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On April 8, 1974 when Hank Aaron surpassed the career home run record of 714 set by the immortal Babe Ruth by hitting his own 715th homer, mathematicians found yet another reason to cheer. You see, if we prime factorize these two numbers, we have the following:
Now a quick examination reveals a unique property: the first seven primes, and only those, are used! That certainly wouldn't happen very often, would it? But a deeper look produces another relationship: the sums of the primes for each number are the same!
and
29 = 5 + 11 + 13
Surprising or spooky, take your pick. But if you are a true number lover, you should be asking yourself this question: "Are there other pairs of consecutive numbers that behave in this way?" According to Paul Hoffman's book, "The Man Who Loved Only Numbers: The Story of Paul Erdös and the Search for Mathematical Truth"[1], Carl Pomerance did. And the answer was in the affirmative. However, as might be suspected, such Ruth-Aaron pairs, as they can be called, appear somewhat infrequently. He did a computer search of all the positive integers less than 20,000 and found only 26 pairs. The smallest pair is 5 and 6; the largest, 18,490 and 18491. [2]
Then, in the tradition of all good number theorists, he conjectured (but did not prove) that the number of such pairs was infinite. That is where Paul Erdös entered the picture, setting to rest the dilemma; again the answer was in the affirmative.
Where do we go from here? Is this just a little tidbit of mathematical whimsy, or can it be turned into something worthwhile, at least for students and teachers of school math? We believe it can, and here is how we do it.
Initially, the students must have a working background of two ideas: prime numbers and the prime factorization of numbers. Once that is established, the students can be presented with the topic of Ruth-Aaron numbers, starting off with the story about the home run records of the two famous baseball greats. This is followed up with mention of what Pomerance did to search for more instances. We usually tell the class that 5 and 6 is the smallest pair, showing why it is so. Then we suggest they start looking for more pairs themselves. At first, shock sets in; they don't quite know what to make of it. It's a good idea to assure them that the next two pairs are rather small.
After they have been found -- (8, 9) and (15, 16) -- we go for the really big show and give them the largest pair in Pomerance's search, asking them to prove that it is in fact an R-A pair. With calculators at hand and basic knowledge of the divisibility tests normally taught in middle school math, the request is a reasonable one. All the primes necessary for the task are less than 50 and the sum of the prime sets is less than 100. 'Nuff said.
Ted Alper, of the Education Program for Gifted Youth at Stanford University, has provided us with the complete list of R-A pairs for numbers less than 1,000,000. [3] By his count, there are 149 such pairs. Therefore, we can do many additional exercises in any of three ways: (1) present a specific pair and ask that it be proved by showing the prime factorizations and the equal sums; or (2) present a set of four or five number pairs (only some of which are legitimate cases) and say, "Which pairs given here are indeed R-A numbers?", or (3) state that an R-A pair exists in a certain small range of numbers, say a decade, then ask the students to find which pair of consecutive numbers is the one we seek. An example of the last approach might be: find the R-A pair hidden in the range of numbers from 5400 to 5410.
But why let a good thing stop there? A good mathematician, like a good musician or artist, is always creating new variations on a basic theme. Here is one such variation among other possibilities: find a pair of consecutive numbers for which their RAV's [i.e. Ruth-Aaron Value, or sum of the primes] are themselves consecutive. A simple example of this is 24 and 25. The RAV(24) = 2 + 2 + 2 + 3 = 9 and RAV(25) = 5 + 5 = 10. Many such pairs of this type exist, of course. According to Alper, there are 55 such pairs less than 100,000. But, there is a catch here. Those pairs are what we call "Type i", where "i" is for "increasing". This can easily be expressed using "function notation", as follows:
This implies there must be a second category, "Type d", where "d" is for "decreasing". The pair of 14 and 15 serves as a simple illustration; RAV(14) = 9 and RAV(15) = 8. Again using functions, we have
Once again, Alper's computer reveals that there are 51 such pairs less than 100,000 in this category. [4]
We believe the use of function notation provides a unique way to present this topic in an advanced math manner. And therefore justify its inclusion in school math lessons. It provides a new challenge to the students while preparing them for their future study of functions in algebra and beyond.
Additional ideas for investigation include the following:
(1) Do there exist pairs of consecutive numbers for which for some positive integer k > 1,
[Put in middle school math terms, this merely says: "The RAV of one number is a multiple of the RAV of the other number."] The answer is yes, as might be expected. There are 43 such pairs in the range of numbers less than 2000. And for the record, 222 pairs less than 20,000; 630 pairs less than 100,000; and 2927 pairs less than 1,000,000. [4]
(2) Do there exist numbers n such that n = kRAV(n) for some positive integer k > 1? This means the RAV of a number is a factor (or divisor) of the given number. An example: RAV(60) = 15 and 15 is a factor of 60. [Note: this is a trivial case when n is prime, because RAV(n) = n. So non-prime values of n are what we are seeking here.]
(3) Take a number and double it. Notice that the RAV of the doubled number is only 2 more than the RAV of the first number. Can you explain this? For example the RAV(12) = 7, but the double of 12 is 24 and the RAV(24) = 9, which is only 2 more.
(4) Observe that RAV(291) = 100, the "century" number. It is the smallest number with this property. What is the next largest number whose RAV is 100?
(5) Define a recursive relationship on the set of whole numbers as follows: For a given n, find RAV(n). Then find the RAV of that result. Continue this process of finding the RAV of each succeeding outcome, until... Well, what do you think will happen? We call this the longevity of the original number n.
An example is 91. RAV(91) = 20.
RAV(20) = 9.
RAV(9) = 6.
RAV(6) = 5.
RAV(5) = 5 DEAD-END!
The process terminates with 5, because 5 is a prime.
This means 91 --> 20 --> 9 --> 6 --> 5. Therefore, the longevity of 91 is 4. [We might symbolize this by writing RAV4(91) = 5 and L(91) = 4.]
Numbers can now be categorized by (1) their longevity numbers, or (2) according to which primes they eventually arrive. [For a more in-depth discussion of this feature, see [5].]
It seems that the possible avenues for investigation are almost endless, or at least, certainly varied enough to encourage individual creativity by anyone willing to try something new.
[1] Hoffman, Paul. "The Man Who Loved Only Numbers: The Story of Paul Erdös and the Search for Mathematical Truth". Hyperion: New York. 1998.
[3] Alper, Ted. E-mail, May 29, 1999.
[4] Alper, Ted. E-mail, June 10, 1999.
[5] Dane, Perry, "The 'Prime Derivative'", Journal of Recreational Mathematics, Vol. 7, No. 2 (1974), p. 111-115.
For a different slant on home runs in baseball with mathematics, go to Mike Keith's site on Maris-McGwire-Sosa Numbers.
For a discussion of Ruth-Aaron Triplets in The Prime Puzzles and Problems Connection, click HERE.
For more useful information about generating RA Pairs, go to Joe K. Crump's Number Theory Web.
1. The first ten cases of regular R-A pairs:
Pair 1: (5,6) with sum 5. Pair 2: (8,9) with sum 6. Pair 3: (15,16) with sum 8. Pair 4: (77,78) with sum 18. Pair 5: (125,126) with sum 15. Pair 6: (714,715) with sum 29. Pair 7: (948,949) with sum 86. Pair 8: (1330,1331) with sum 33. Pair 9: (1520,1521) with sum 32. Pair 10: (1862,1863) with sum 35.
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2. Alper also considered the variation on the basic theme of merely finding consecutive pairs for which sums of the distinct primes in each prime factorization were used. Here are the first cases:
Pair 1: (5,6) with sum 5. Pair 2: (24,25) with sum 5. Pair 3: (49,50) with sum 7. Pair 4: (77,78) with sum 18. Pair 5: (104,105) with sum 15. Pair 6: (153,154) with sum 20. Pair 7: (369,370) with sum 44. Pair 8: (492,493) with sum 46. Pair 9: (714,715) with sum 29. Pair 10: (1682,1683) with sum 31.
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3. This list is for "Type i" pairs as explained in the article.
Pair 1: (2,3) with first sum 2. Pair 2: (3,4) with first sum 3. Pair 3: (4,5) with first sum 4. Pair 4: (9,10) with first sum 6. Pair 5: (20,21) with first sum 9. Pair 6: (24,25) with first sum 9. Pair 7: (98,99) with first sum 16. Pair 8: (170,171) with first sum 24. Pair 9: (1104,1105) with first sum 34. Pair 10: (1274,1275) with first sum 29.
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4. This list is for "Type d" pairs as explained in the article.
Pair 1: (7,8) with sums (7,6). Pair 2: (14,15) with sums (9,8). Pair 3: (63,64) with sums (13,12). Pair 4: (80,81) with sums (13,12). Pair 5: (224,225) with sums (17,16). Pair 6: (285,286) with sums (27,26). Pair 7: (351,352) with sums (22,21). Pair 8: (363,364) with sums (25,24). Pair 9: (475,476) with sums (29,28). Pair 10: (860,861) with sums (52,51).
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5. List for RAVs in which one is a non-trivial multiple of the other.
Pair 1: (74,75) with sums (39,13). Pair 2: (160,161) with sums (15,30). Pair 3: (174,175) with sums (34,17). Pair 4: (252,253) with sums (17,34). Pair 5: (259,260) with sums (44,22). Pair 6: (287,288) with sums (48,16). Pair 7: (335,336) with sums (72,18). Pair 8: (391,392) with sums (40,20). Pair 9: (395,396) with sums (84,21). Pair 10: (447,448) with sums (152,19).
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6. List of the first cases where the RAV is a factor of the given number.
Value 1: 16 has RAV 8. Value 2: 27 has RAV 9. Value 3: 30 has RAV 10. Value 4: 60 has RAV 12. Value 5: 70 has RAV 14. Value 6: 72 has RAV 12. Value 7: 84 has RAV 14. Value 8: 105 has RAV 15. Value 9: 150 has RAV 15. Value 10: 180 has RAV 15.
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