Ɉ ȽɂɉɈɌȿɁȿ ɄɈɅɅȺɌɐȺ Ⱥɦɨɫɨɜ ȿɜɝɟɧɢɣ Ⱥɥɟɤɫɚɧɞɪɨɜɢɱ

ɆȺɌȿɆȺɌɂɄȺ

Ɉ ȽɂɉɈɌȿɁȿ ɄɈɅɅȺɌɐȺ

Ⱥɦɨɫɨɜ ȿɜɝɟɧɢɣ Ⱥɥɟɤɫɚɧɞɪɨɜɢɱ

ɞɨɰ., ɋɚɦɚɪɫɤɢɣ ɝɨɫɭɞɚɪɫɬɜɟɧɧɵɣ ɬɟɯɧɢɱɟɫɤɢɣ ɭɧɢɜɟɪɫɢɬɟɬ, ɊɎ, ɝ. ɋɚɦɚɪɚ

ABOUT COLLATZ CONJECTURE

Evgeniy Amosov associate professor, Samara state technical university Russia, Samara

ȺɇɇɈɌȺɐɂə

ɉɨɤɚɡɚɧɨ, ɱɬɨ ɫ ɩɨɦɨɳɶɸ ɨɩɟɪɚɰɢɣ ɞɟɥɟɧɢɹ ɧɚ 2 ɢ ɭɦɧɨɠɟɧɢɹ ɧɚ 3 ɫ ɩɪɢɛɚɜɥɟɧɢɟɦ 1 ɦɨɠɧɨ ɩɨɥɭɱɢɬɶ ɰɟɥɨɟ ɱɢɫɥɨ, ɤɪɚɬɧɨɟ 5. ABSTRACT

It is shown that using the operations of division by 2 and multiplication by 3 with the addition of 1, you can get an integer multiple of 5.

Ʉɥɸɱɟɜɵɟ ɫɥɨɜɚ: ɝɢɩɨɬɟɡɚ Ʉɨɥɥɚɬɰɚ, ɫɢɪɚɤɭɡɫɤɚɹ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ.

Keywords: Collatz Conjecture, Syracuse sequence.

ȼɨɡɶɦёɦ ɥɸɛɨɟ ɧɚɬɭɪɚɥɶɧɨɟ ɱɢɫɥɨ n. ȿɫɥɢ ɨɧɨ ɱёɬɧɨɟ, ɬɨ ɟɝɨ ɫɥɟɞɭɟɬ ɪɚɡɞɟɥɢɬɶ ɧɚ 2, ɚ ɟɫɥɢ ɧɟɱёɬɧɨɟ, ɬɨ ɭɦɧɨɠɢɬɶ ɧɚ 3 ɢ ɩɪɢɛɚɜɢɬɶ 1 (ɜ ɢɬɨɝɟ ɩɨɥɭɱɚɟɬɫɹ 3n + 1). ɇɚɞ ɩɨɥɭɱɟɧɧɵɦ ɱɢɫɥɨɦ ɜɵɩɨɥɧɹɸɬɫɹ ɬɟ ɠɟ ɫɚɦɵɟ ɞɟɣɫɬɜɢɹ, ɢ ɬɚɤ ɞɚɥɟɟ.

ȽɢɩɨɬɟɡɚɄɨɥɥɚɬɰɚ ɡɚɤɥɸɱɚɟɬɫɹ ɜ ɬɨɦ, ɱɬɨ ɤɚɤɨɟ ɛɵ ɧɚɱɚɥɶɧɨɟ ɱɢɫɥɨ n ɧɢ ɛɵɥɨ ɜɡɹɬɨ, ɪɚɧɨ ɢɥɢ ɩɨɡɞɧɨ ɩɨɥɭɱɢɬɫɹ ɟɞɢɧɢɰɚ [1, 2]. ɉɨɤɚɠɟɦ, ɱɬɨ ɜ ɪɟɡɭɥɶɬɚɬɟ ɞɟɣɫɬɜɢɹ ɭɤɚɡɚɧɧɵɯ ɞɜɭɯ ɨɩɟɪɚɰɢɣ (ɞɟɥɟɧɢɹ ɧɚ 2 ɢ ɭɦɧɨɠɟɧɢɹ ɧɚ 3 ɫ ɩɪɢɛɚɜɥɟɧɢɟɦ 1), ɜ ɤɨɧɰɟ ɤɨɧɰɨɜ ɦɵ ɩɨɥɭɱɢɦ ɱɢɫɥɨ, ɤɨɬɨɪɨɟ ɞɟɥɢɬɫɹ ɧɚ 5. ɉɪɟɞɫɬɚɜɢɦ ɱɢɫɥɨ n ɜ ɫɥɟɞɭɸɳɟɦ ɜɢɞɟ n=10a + b, ɝɞɟ a –ɰɟɥɨɟ ɱɢɫɥɨ, ɚ b –ɧɚɬɭɪɚɥɶɧɨɟ ɱɢɫɥɨ, ɤɨɬɨɪɨɟ ɦɨɠɟɬ ɩɪɢɧɢɦɚɬɶ ɡɧɚɱɟɧɢɟ ɨɬ 1 ɞɨ 9. Ɍɚɤ ɤɚɤ ɱёɬɧɨɟ ɱɢɫɥɨ ɞɨɥɠɧɨ ɞɟɥɢɬɶɫɹ ɧɚ 2, ɬɨ ɛɭɞɟɦ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɬɨɥɶɤɨ ɧɟɱёɬɧɵɟ ɱɢɫɥɚ b, ɬɨɟɫɬɶ, b=1, 3, 5, 7, 9.

Ɉɱɟɜɢɞɧɨ, ɱɬɨ ɜ ɫɥɭɱɚɟ b=5, ɱɢɫɥɨ n ɞɟɥɢɬɫɹ ɧɚ 5 ɩɨ ɩɪɢɡɧɚɤɭ ɞɟɥɢɦɨɫɬɢ ɧɚ 5.

ȿɫɥɢ

b=3, ɱɢɫɥɨ n ɛɭɞɟɬ ɧɟɱёɬɧɵɦ, ɩɨɷɬɨɦɭ ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɫɥɟɞɭɸɳɢɟ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ 3∙3+1=10, ɬɨ ɟɫɬɶ, ɩɨɥɭɱɢɥɨɫɶ ɱɢɫɥɨ, ɤɨɬɨɪɨɟ ɞɟɥɢɬɫɹ ɧɚ 5. ȼ ɫɥɭɱɚɟ b=1 ɡɚɩɢɲɟɦ ɪɚɜɟɧɫɬɜɚ 1∙3+1=4, 4∙3+1=13=10+3, 3∙3+1=10, ɬɨ ɟɫɬɶ, ɷɬɨɬ ɫɥɭɱɚɣ ɫɜɨɞɢɬɫɹ ɤ ɩɪɟɞɵɞɭɳɟɦɭ, ɢɥɢ 4/2=2, 2∙3+1=7, ɢ ɦɵ ɩɨɥɭɱɚɟɦ ɫɥɟɞɭɸɳɢɣ ɫɥɭɱɚɣ. Ⱦɥɹ ɱɢɫɥɚ b, ɪɚɜɧɨɝɨ b=7 ɩɨɥɭɱɚɟɦ 7∙3+1=22, 22/2 =11=10+1, ɱɬɨ ɬɨɠɟ ɫɜɨɞɢɬɫɹ ɤ ɪɚɫɫɦɨɬɪɟɧɧɨɦɭ ɪɚɧɟɟ ɫɥɭɱɚɸ. ɂ, ɧɚɤɨɧɟɰ, ɩɪɢ b=9 ɛɭɞɟɦ ɢɦɟɬɶ 9∙3+1=28, 28/2 =14=10+4, 4∙3+1=13=10+3, 3∙3+1=10.

Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɞɟɣɫɬɜɢɬɟɥɶɧɨ, ɢɡ ɜɵɲɟɢɡɥɨɠɟɧɧɨɝɨ ɜɵɬɟɤɚɟɬ, ɱɬɨ, ɩɪɢɦɟɧɹɹ ɞɜɟ ɭɤɚɡɚɧɧɵɟ ɨɩɟɪɚɰɢɢ, ɦɵ ɛɭɞɟɦ ɩɨɥɭɱɚɬɶ ɱɢɫɥɚ, ɤɨɬɨɪɵɟ ɞɟɥɹɬɫɹ ɧɚɰɟɥɨ ɧɚ 5, ɢɧɚɱɟ ɝɨɜɨɪɹ, ɨɧɢ ɦɨɝɭɬ ɛɵɬɶ ɩɪɟɞɫɬɚɜɥɟɧɵ ɜ ɜɢɞɟ 5k. ɋɥɟɞɭɟɬ ɨɬɦɟɬɢɬɶ, ɱɬɨ ɟɫɥɢ ɧɟɱёɬɧɨɟ ɱɢɫɥɨ ɨɤɚɧɱɢɜɚɟɬɫɹ ɧɚ 1, ɬɨ ɩɨɫɥɟ ɭɦɧɨɠɟɧɢɹ ɧɚ 3 ɢ ɩɪɢɛɚɜɥɟɧɢɹ 1 ɦɨɠɧɨ ɩɨɥɭɱɢɬɶ ɱɢɫɥɨ, ɹɜɥɹɸɳɟɟɫɹ ɫɬɟɩɟɧɶɸ ɱɢɫɥɚ 2. ȼ ɷɬɨɦ ɫɥɭɱɚɟ, ɨɱɟɜɢɞɧɨ, ɩɭɬёɦ ɦɧɨɝɨɤɪɚɬɧɨɝɨ ɞɟɥɟɧɢɹ ɩɨɥɭɱɟɧɧɨɝɨ ɱɢɫɥɚ ɧɚ 2 ɦɵ ɜ ɢɬɨɝɟ ɞɨɣɞёɦ ɞɨ 1, ɤɚɤ ɢ ɭɬɜɟɪɠɞɚɟɬ ɝɢɩɨɬɟɡɚ Ʉɨɥɥɚɬɰɚ. ɉɨɤɚɠɟɦ ɬɟɩɟɪɶ, ɱɬɨ ɫ ɩɨɦɨɳɶɸ ɭɤɚɡɚɧɧɵɯ ɨɩɟɪɚɰɢɣ ɜ ɝɢɩɨɬɟɡɟ Ʉɨɥɥɚɬɰɚ ɦɵ ɦɨɠɟɦ ɩɨɥɭɱɢɬɶ ɱɢɫɥɚ, ɤɨɬɨɪɵɟ ɛɭɞɭɬ ɞɟɥɢɬɶɫɹ ɧɚ 2. Ɍɚɤ ɤɚɤ ɱɢɫɥɨ, ɤɪɚɬɧɨɟ 5, ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɤɚɤ 5k=4k+k, ɝɞɟ ɱɢɫɥɨ k, ɧɚɩɪɢɦɟɪ, ɧɟɱёɬɧɨɟ, ɬɨ ɦɵ ɫɦɨɠɟɦ ɩɪɟɜɪɚɳɚɬɶ ɟɝɨ ɜ ɱёɬɧɨɟ ɱɢɫɥɨ, ɚ ɞɟɥɹ ɩɨɥɭɱɟɧɧɨɟ ɱёɬɧɨɟ ɱɢɫɥɨ ɧɚ 2, ɜ ɤɨɧɰɟ ɤɨɧɰɨɜ ɫɦɨɠɟɦ ɩɨɥɭɱɢɬɶ 1, ɱɬɨ ɢ ɛɭɞɟɬ ɩɨɞɬɜɟɪɠɞɟɧɢɟɦ ɝɢɩɨɬɟɡɵ Ʉɨɥɥɚɬɰɚ.

ȿɫɥɢ ɱɢɫɥɨ k –ɧɟɱёɬɧɨɟ, ɬɨ ɟɝɨ ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɤɚɤ

k=4l+1, k=4l+3, ɝɞɟ l –ɧɟɤɨɬɨɪɨɟ ɰɟɥɨɟ ɱɢɫɥɨ, ɨɱɟɜɢɞɧɨ, ɦɟɧɶɲɟɟ ɱɢɫɥɚ k. ȼ ɩɟɪɜɨɦ ɫɥɭɱɚɟ ɡɚɩɢɲɟɦ (4l+1) ∙3+1=12l+4, ɢ ɭ ɧɚɫ ɩɨɥɭɱɚɟɬɫɹ ɱɢɫɥɨ, ɤɨɬɨɪɨɟ ɞɟɥɢɬɫɹ ɧɚ 4. ȼɨ ɜɬɨɪɨɦ ɫɥɭɱɚɟ (4l+3) ∙3+1=12l+10=(12l+8) +2. ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɩɨɥɭɱɢɥɨɫɶ ɱёɬɧɨɟ ɱɢɫɥɨ ɜɢɞɚ 4m+2, ɬɚɤ ɤɚɤ ɜɵɪɚɠɟɧɢɟ ɜ ɫɤɨɛɤɚɯ, ɨɱɟɜɢɞɧɨ, ɞɟɥɢɬɫɹ ɧɚ 4.

Ɉɬɦɟɬɢɦ, ɱɬɨ ɩɪɢ ɞɟɥɟɧɢɢ ɱɢɫɥɚ ɜɢɞɚ 4m+2 ɧɚ 2 ɦɵ ɩɨɥɭɱɢɦ ɧɟɱёɬɧɨɟ ɱɢɫɥɨ ɜɢɞɚ 2m+1, ɩɪɟɨɛɪɚɡɨɜɚɧɢɟ ɤɨɬɨɪɨɝɨ ɛɭɞɟɬ ɜɵɝɥɹɞɟɬɶ ɬɚɤ (2m+1) ∙3+1=6m+4, (6m+4)/2 =3m+2. ɇɟɱёɬɧɨɟ ɱɢɫɥɨ 3m+2 ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɜ ɨɞɧɨɦ ɢɡ ɫɥɟɞɭɸɳɢɯ ɜɢɞɨɜ 3m+2=4z+1, 3m+2=4z+3, ɝɞɟ z- ɧɟɤɨɬɨɪɨɟ ɰɟɥɨɟ ɱɢɫɥɨ, ɬɨ ɟɫɬɶ, ɫɜɟɫɬɢ ɡɚɞɚɱɭ ɤ ɪɚɫɫɦɨɬɪɟɧɧɨɦɭ ɪɚɧɟɟ ɫɥɭɱɚɸ.

Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɢɡ ɧɚɲɢɯ ɪɚɫɫɭɠɞɟɧɢɣ ɦɨɠɧɨ ɫɞɟɥɚɬɶ ɜɵɜɨɞ, ɱɬɨ ɩɭɬёɦ ɞɜɭɯ ɜɢɞɨɜ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ ɦɵ ɛɭɞɟɦ ɩɨɥɭɱɚɬɶ ɱɢɫɥɚ, ɤɪɚɬɧɵɟ ɱɢɫɥɭ 5 ɢ ɱёɬɧɵɟ ɱɢɫɥɚ. ɗɬɨ ɩɨɡɜɨɥɹɟɬ ɧɚɦ ɩɪɢɣɬɢ ɤ ɡɚɤɥɸɱɟɧɢɸ ɨ ɬɨɦ, ɱɬɨ ɫ ɩɭɬёɦ ɩɨɜɬɨɪɟɧɢɹ ɞɚɧɧɵɯ ɨɩɟɪɚɰɢɣ (ɞɟɥɟɧɢɹ ɧɚ 2 ɢ ɭɦɧɨɠɟɧɢɹ ɧɚ 3 ɫ ɩɪɢɛɚɜɥɟɧɢɟɦ 1), ɦɵ ɩɨɥɭɱɢɦ ɜ ɢɬɨɝɟ ɥɢɛɨ 5, ɥɢɛɨ 1. ȼ ɫɥɭɱɚɟ ɱɢɫɥɚ 5 ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ 3∙5+1=16, 16/2=8, 8/2=4, 4/2=2. 2/2=1.

Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɜ ɫɥɭɱɚɟ 5 ɦɵ ɬɨɠɟ ɜ ɢɬɨɝɟ ɩɪɢɯɨɞɢɦ ɤ ɟɞɢɧɢɰɟ. ȼɵɜɨɞ. ɉɪɢɜɟɞɟɧɵ ɚɪɝɭɦɟɧɬɵ ɜ ɩɨɥɶɡɭ ɜɟɪɧɨɫɬɢ ɝɢɩɨɬɟɡɵ Ʉɨɥɥɚɬɰɚ ɨ ɬɨɦ, ɱɬɨ ɤɚɤɨɟ ɛɵ ɧɚɱɚɥɶɧɨɟ ɱɢɫɥɨ n ɧɢ ɛɵɥɨ ɜɡɹɬɨ, ɪɚɧɨ ɢɥɢ ɩɨɡɞɧɨ ɩɨɥɭɱɢɬɫɹ ɟɞɢɧɢɰɚ, ɟɫɥɢ ɦɵ ɛɭɞɟɦ ɩɪɢɦɟɧɹɬɶ 2 ɨɩɟɪɚɰɢɢ: ɞɟɥɟɧɢɟ ɱёɬɧɨɝɨ ɱɢɫɥɚ ɧɚ 2 ɢ ɭɦɧɨɠɟɧɢɟ ɧɟɱёɬɧɨɝɨ ɱɢɫɥɚ ɧɚ 3 ɫ ɩɪɢɛɚɜɥɟɧɢɟɦ 1.

ɋɩɢɫɨɤɥɢɬɟɪɚɬɭɪɵ:

1. Lagarias J. The 3x+1 problem and its generalizations // American Mathematical Monthly. — 1985. — Vol. 92. —

P. 3—23. 2. ɋɬɸɚɪɬ ɂ. ȼɟɥɢɱɚɣɲɢɟ ɦɚɬɟɦɚɬɢɱɟɫɤɢɟɡɚɞɚɱɢ. —Ɇ.: Ⱥɥɶɩɢɧɚ ɧɨɧ-ɮɢɤɲɧ, 2015. —460 ɫ. 3. ɏɷɣɟɫ Ȼ. ȼɡɥёɬɵ ɢ ɩɚɞɟɧɢɹ ɱɢɫɟɥ-ɝɪɚɞɢɧ // ȼ ɦɢɪɟ ɧɚɭɤɢ (Scientific American). — 1984. —№ 3. —ɋ. 102— 107. 4. Winkler. Ɋ. Mathematical Puzzles: A connoisseur's collection // The Mathematical Gazette. – 2006. – 90(517). –

Ɋ.186-188.