IF(A2-$A$2 + 1 < =$AHS2,SMALL($AES2:AES501,A2-SA$2 + l),0) and replicate it to cell AF100. This makes it possible to set up primes in column. AF one after ...
!"#$%&'(#)*+%,&$%-.,/*.&-+-%012 3/2%4%5&/6&'1%7898 :))+$$%-+(&#*$2%:))+$$%;+(&#*$2%0, AF3—AF2," ") is denned in cell AJ3 and replicated to cell AJ100. Then in cell AK2 the spreadsheet function = MODE(AJ3:AJ100) is defined. As a result numbers in column AJ show all possible gaps and a number in cell AK2 indicates the smallest most common gap within the given range (for example, in the interval (6500,7000) gaps 2 and 6 both occur 12 times whereas cell AK2 displays the gap 2). 5. Primes in arithmetic progression A classical and most important result about primes in arithmetic progression states: If d ^ 2 and « ^ 0 are integers that are relatively prime, than the arithmetic progression a, a + d, a + 2d, a + 3d, . . . consists of infinitely many primes. This theorem was proved by Dirichiet in 1837. The proof requires more than elementary means. In this respect we agree with Stewart who noted that 'the fact
Downloaded At: 13:18 8 January 2010
Exploring prime numbers within a spreadsheet
697
that proof is important for the professional mathematician does not imply that the teaching of mathematics to a given audience must be limited to ideas whose proofs are accessible to that audiences' [8]. Indeed, the students can use a spreadsheet to verify this theorem for different values of a and d. It may be of interest to find how many times the same gap occurs between consecutive pairs of primes. For example, a gap of size 6 occurs three times running in sequences 251, 257, 263, 269, and 3301, 3307, 3313, 3319. A gap of size 12 occurs twice between three consecutive primes 4679, 4691, 4703. There exist five sequential primes in arithmetic progression with 6 as a common difference: 5, 11, 17, 23, 29. Is it possible to find another five primes in arithmetic progression with this property? One more activity might be in exploring the arithmetic progression 199 + 210& which for k = 0, 1, 2, . . ., 9 consists of ten primes. A spreadsheet allows the students to do this kind of explorations. Another interesting investigation is a search for primes with gaps in arithmetic progression. Students could be demonstrated that primes 347, 349, 353, 359, 367, and 4931, 4933, 4937, 4943, 4951 have gaps in arithmetic progression, and then they could be encouraged to look for other examples by simply changing entries of cell A2 in the spreadsheet. 6. Exploring the prime counting function The use of a spreadsheet allows students to visualize at once the behaviour of the function n(x) which represents a number of primes less than x. In order to make possible the changes in values of the function n(x) as a response to any new entry of cell A2, the students can be encouraged to count the number of primes within each range 2:499, 500:999, . . ., 10000:10499, and then sum these values subsequently in order to get the number of primes occurring prior to a range under consideration. For example, 168 is the number of primes occurring prior to the range 1000:1499. This makes it possible to define in cell AL3 and then replicate to cell AL501 the following spreadsheet function = IF(SAS2 = 2,COUNT(SAE$2:$AE3),0) + IF($A$2 = 500,COUNT(SAE$2:$AE3) + 95,0) + IF($A$2 = 1000,COUNT($AE$2:$AE3) +168,0) + IF($A$2 = 1500,COUNT($AE$2:$AE3) + 239,0) + IF($A$2 = 2000,COUNT($AE$2:$AE3) + 303,0) + IF($A$2 = 2500,COUNT(SAE$2:SAE3) + 367,0) + IF($A$2 = 3000,COUNT($AE$2:SAE3) + 430,0) + IF($A$2 = 3500,COUNT($AE$2:$AE3) + 489,0) + IF(SA$2=4000,COUNT($AES2:$AE3) + 550,0) + IF($A$2 = 4500,COUNT($AE$2:SAE3) + 610,0) + IF($A$2 = 5000,COUNT($AE$2:$AE3) + 669,0) + IF($AS2 = 5500,COUNT($AE$2:$AE3) +725,0) + IF(SA$2 = 6000,COUNT($AES2:$AE3)+ 783,0) + IF($A$2 = 6500,COUNT(SAE$2:$AE3) + 842,0) + IF($A$2 = 7000,COUNT($AE$2:$AE3) + 900,0) + IF($AS2 = 7500,COUNT($AE$2:$AE3) + 950,0) + IF($A$2 = 8000,COUNT($AES2:$AE3) +1007,0) + IF($A$2 = 8500,COUNT($AE$2:$AE3) +1059,0) +
698
. Abramovich
Downloaded At: 13:18 8 January 2010
Figure 3. The graph of (n(x)\n(x))/x on the interval (4000, 4499). IF(SA$2 = 9000,COUNT($AE$2:$AE3) + 1117,0) + IF($AS2 = 9500,COUNT(SAES2:SAE3) +1177,0) + IF($AS2 = 10000,COUNT($AE$2:SAE3) + 1229,0). This function calculates values of n(x) by using one of the summands in dependency on the entry of cell A2. Finally, the spreadsheet function =AL3*LN(A3)/A3 can be denned in cell AM3 and replicated down to cell AM501. This, along with spreadsheet graphics allows one to display both in numerical and graphical form the celebrated prime number theorem. The graph of the ratio n(x)\n(x)/x on the interval (4000, 4499) is shown in Figure 3. It should be noted that this graph can be changed instantly by a simple variation of an input value in cell A2. 7. Conclusions This article demonstrates the use of computer spreadsheet in exploring aspects of the theory of numbers. Less than 10 spreadsheet functions allow students to create a powerful and flexible learning medium which has many advantages in comparison with the time-consuming students' exposure to BASIC. For example, a computer program presented in [3] which performs similar investigations, apart from the lack of visualization, consists of 10 times as many BASIC operators. Spreadsheet explorations are not limited to the range of 10000 integers — the extension of prime numbers in row 1 by some 15 primes results in expanding the range of numbers to be explored by a factor of 10. It is also important to recognize that flexibility of a spreadsheet in the context of independent explorations creates an open interactive learning environment allowing individual constructions of situations based on individual interest. All these actually open up the possibility of making the study of number theory an exciting challenge for independent investigation.
Exploring prime numbers within a spreadsheet
699
Acknowledgements The author is grateful to James Wilson, and Lyle Pagnucco for their valuable comments on earlier drafts.
References [1]
SPENCER,
[2] [3] [4]
OWENS, J. E., 1990, J. Comput. Math. Sci. Teaching, 10, 67. PETERSON, G. D., 1994, Math. Comput. Educ, 28, 154. ABRAMOVICH, S., FUJII, T., and WILSON, J., 1994, Proceedings of
Downloaded At: 13:18 8 January 2010
[5] [6] [7] [8]
Press).
D. D., 1982, Computers in Number Theory (Rockville, MD: Computer Science
the 1994 International Symposium on Mathematics/Science Education and Technology, edited by G. H. Marks (Charlottesville, VA: Association for the Advancement of Computing in Education) p. 1. STEWARD, T., 1994, Int. J. Math. Educ. Sci. Technol., 25, 239. MACKINNON, N., 1993, Math. Gazette, 77(478), 2. ABRAMOVICH, S., 1995, Int. J. Math. Educ. Sci. TechnoL, 26, 347. STEWART, I., 1990, On the Shoulders of Giants. New Approaches to Numeracy, edited by L. A. Steen (Washington, DC: National Academy Press) pp. 183-217.
