Problems & Puzzles: Puzzles Puzzle 65. Multigrade Relations 1 + 6 + 8 = 2 + 4
+ 9 & is the least example of the socalled multigrade relations. It's common to express them the following way: 1, 6, 8 = 2, 4, 9; for n=1, 2 Here are others reported: 1, 5, 8, 12 = 2, 3, 10, 11; for n=1, 2, 3 There are also published in the literature, multigrade relations involving only prime numbers: 43, 61, 67 = 47, 53, 71; for n=1, 2 Results from my own search up today are: A) Multigradeprimerelations formed by 6 distinct primes, 3 each side, for n = 1 &2
B) Multigradeprimerelations with 8 distinct primes, 4 by side, for n = 1, 2 & 3: Using the known multigrade relation: 1, 79, 105, 183 = 3, 69, 115, 181 = 13, 45, 139, 171, for n = 1, 2 & 3 I have found that adding 58, to each term we get: 59, 137, 163, 241 = 61, 127, 173, 239 = 71, 103, 197, 219, for n = 1, 2 & 3. C) Multigradeprimerelations with 8 primes, 4 by side, using 6 consecutive primes, for n = 1, 2 & 3: Using the known Multigrade relation: 5, 57, 127, 177 = 27, 27, 157, 157, for n = 1, 2 & 3 I have obtained the following one with 6 consecutive primes: 91335911, 91335961, 91336031, 91336081 = 91335931, 91335931, 91336061, 91336061, for n = 1, 2, 3 D) Multigradeprimerelations with 10 members, 5 by side, for n=1, 3, 5 & 7 (17, 71, 103, 157, 163 = 31, 47, 121, 143, 169) adding 210 to each term, for n = 1, 3, 5 & 7. (17, 71, 103, 157, 163 = 31, 47, 121, 143, 169) adding 1830 to each term, for n = 1, 3, 5 & 7. Would you like to find other / larger / more interesting multigradeprimerelations? References: 1.Recreations in The Theory of Numbers, Albert
H. Beiler, p. 163 Solutions T.W.A. Baumann found (31/08/99) the following pentagradeprimerelation: 277, 937, 1069, 2389,
2521, 3181 = 409, 541, 1597, 1861, 2917, 3049 This is his method "I used the (known)
integer relation: The same way but now using the known heptagrade
relation: 12251 13841 n.b. First column is left side of the relation; second column is the right side of the relation. *** On Feb 2014 Dr. Jonathan Sondow wrote:
A few days later he added:
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