Ɉ ɊȺɁȻɂȿɇɂɂ ɇȿɑЁɌɇɕɏ ɑɂɋȿɅ ɇȺ ɋɅȺȽȺȿɆɕȿ ɋ ɍɑȺɋɌɂȿɆ ɉɊɈɋɌɕɏ ɑɂɋȿɅ Ⱥɦɨɫɨɜ ȿɜɝɟɧɢɣ Ⱥɥɟɤɫɚɧɞɪɨɜɢɱ

№ 31 (160), 2020 ɝ.

Ɉ ɊȺɁȻɂȿɇɂɂ ɇȿɑЁɌɇɕɏ ɑɂɋȿɅ ɇȺ ɋɅȺȽȺȿɆɕȿ ɋ ɍɑȺɋɌɂȿɆ ɉɊɈɋɌɕɏ ɑɂɋȿɅ

Ⱥɦɨɫɨɜ ȿɜɝɟɧɢɣ Ⱥɥɟɤɫɚɧɞɪɨɜɢɱ

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

SPLITTING ODD NUMBERS INTO SUMMANDS WITH THE PARTICIPATION OF PRIME NUMBERS

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

ȺɇɇɈɌȺɐɂə

ɉɨɤɚɡɚɧɨ, ɱɬɨ ɥɸɛɨɟ ɧɟɱёɬɧɨɟ ɱɢɫɥɨ, ɧɚɱɢɧɚɹ ɫ 3, ɦɨɠɧɨ ɩɪɟɞɫɬɚɜɢɬɶ ɤɚɤ ɫɭɦɦɭ ɩɪɨɫɬɨɝɨ ɱɢɫɥɚ ɢ ɱɢɫɥɚ, ɤɪɚɬɧɨɝɨ 6. ABSTRACT

It is shown that any odd number starting from 3 can be represented as the sum of a prime number and a multiple of 6.

Ʉɥɸɱɟɜɵɟ ɫɥɨɜɚ: ɩɪɨɫɬɵɟ ɱɢɫɥɚ, ɪɚɡɥɨɠɟɧɢɟ ɧɚ ɫɥɚɝɚɟɦɵɟ. Keywords: prime numbers, decomposition into term.

ȼ ɛɨɥɟɟ ɪɚɧɧɢɯ ɪɚɛɨɬɚɯ ɛɵɥɢ ɪɚɫɫɦɨɬɪɟɧɵ ɜɨɩɪɨɫɵ ɪɚɡɛɢɟɧɢɹ ɱёɬɧɵɯ ɢ ɧɟɱёɬɧɵɯ ɱɢɫɟɥ ɧɚ ɫɥɚɝɚɟɦɵɟ ɜɢɞɚ 6n±1 [1,2]. Ⱦɚɧɧɚɹ ɫɬɚɬɶɹ ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɞɚɥɶɧɟɣɲɟɟ ɢɡɭɱɟɧɢɟ ɫɢɬɭɚɰɢɢ ɪɚɡɥɨɠɟɧɢɹ ɱɢɫɟɥ ɧɚ ɫɥɚɝɚɟɦɵɟ ɫ ɭɱɚɫɬɢɟɦ ɩɪɨɫɬɵɯ ɱɢɫɟɥ. ɉɨɤɚɠɟɦ, ɱɬɨ ɥɸɛɨɟ ɧɟɱёɬɧɨɟ ɱɢɫɥɨ, ɧɚɱɢɧɚɹ ɫ 3, ɦɨɠɧɨ ɩɪɟɞɫɬɚɜɢɬɶ ɤɚɤ ɫɭɦɦɭ ɞɜɭɯ ɱɢɫɟɥ, ɨɞɧɨ ɢɡ ɤɨɬɨɪɵɯ ɩɪɨɫɬɨɟ, ɚ ɜɬɨɪɨɟ ɱɢɫɥɨ ɞɟɥɢɬɫɹ ɧɚ 6 ɛɟɡ ɨɫɬɚɬɤɚ, ɬɨ ɟɫɬɶ, ɦɨɠɟɬ ɛɵɬɶ ɩɪɟɞɫɬɚɜɥɟɧɨ ɤɚɤ 6k, ɝɞɟ k –ɧɟɨɬɪɢɰɚɬɟɥɶɧɨ ɰɟɥɨɟ ɱɢɫɥɨ.

Ⱦɟɣɫɬɜɢɬɟɥɶɧɨ, ɩɪɢɜɟɞёɦ ɩɪɢɦɟɪɵ: Ɉɱɟɜɢɞɧɨ, ɱɬɨ ɩɟɪɜɚɹ ɫɤɨɛɤɚ ɞɟɥɢɬɫɹ ɛɟɡ ɨɫɬɚɬ

3=6∙0+3, 5=6∙0+5, 7=6∙0+7, 9=6∙1+3, 11=6∙1+5, 13=6∙1+7, 15=6∙2+3, 17=6∙2+5.

ɉɨɤɚɠɟɦ, ɱɬɨ ɞɚɧɧɨɟ ɭɬɜɟɪɠɞɟɧɢɟ ɜɟɪɧɨ ɢ ɜ ɨɛɳɟɦ ɫɥɭɱɚɟ. Ʉɚɤ ɛɵɥɨ ɩɨɤɚɡɚɧɨ ɜ [2], ɧɟɱёɬɧɨɟ ȼ ɢɬɨɝɟ ɩɨɥɭɱɚɟɦ ɩɪɟɞɫɬɚɜɥɟɧɢɟ

ɱɢɫɥɨ Nɦɨɠɧɨ ɪɚɡɛɢɬɶ ɧɚ ɬɪɢ ɫɥɚɝɚɟɦɵɯ

N=(6k±1) + (6m±1) + (6l±1),

ɱɢɫɥɚ. Ⱦɨɩɭɫɬɢɦ, ɞɥɹ ɧɚɱɚɥɚ, ɱɬɨ ɭ ɜɫɟɯ ɬɪёɯ ɫɥɚɝɚɟɦɵɯ ɡɧɚɤɢ ɩɥɸɫ. Ɍɨɝɞɚ ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ

N=(6k+6m+6l) + 3.

ɉɟɪɜɨɟ ɫɥɚɝɚɟɦɨɟ, ɨɱɟɜɢɞɧɨ, ɞɟɥɢɬɫɹ ɧɚɰɟɥɨ ɧɚ 6, ɚ ɜɬɨɪɨɟ ɫɥɚɝɚɟɦɨɟ ɹɜɥɹɟɬɫɹ ɩɪɨɫɬɵɦ, ɱɬɨ ɫɨɝɥɚɫɭɟɬɫɹ ɫ ɧɚɲɢɦ ɩɪɟɞɩɨɥɨɠɟɧɢɟɦ. ȿɫɥɢ ɭ ɜɫɟɯ ɬɪёɯ ɫɥɚɝɚɟɦɵɯ ɡɧɚɤɢ ɦɢɧɭɫ, ɬɨ ɫɥɟɞɭɟɬ ɩɪɟɨɛɪɚɡɨɜɚɬɶ N=(6k–1) + (6m–1) + (6l–1) =(6(k–1)+6–1) + (6m–1) + (6l–1), N=(6(k–1)+6m+6l) + 3,

ɢ ɦɵ ɩɨɥɭɱɢɦ ɩɪɟɞɵɞɭɳɢɣ ɫɥɭɱɚɣ. ɉɭɫɬɶ ɬɟɩɟɪɶ ɡɧɚɤɢ ɪɚɡɧɵɟ, ɧɚɩɪɢɦɟɪ, ɞɥɹ ɨɩɪɟɞɟɥёɧɧɨɫɬɢ, ɜ ɩɟɪɜɨɣ ɫɤɨɛɤɟ ɪɚɡɥɨɠɟɧɢɹ ɫɬɨɢɬ ɡɧɚɤ ɩɥɸɫ. Ɍɨɝɞɚ ɢɦɟɟɦ

N=((6k+1) + (6m–1)) + (6l±1).

ɤɚ ɧɚ 6, ɬɚɤ ɤɚɤ ɞɜɟ ɟɞɢɧɢɰɵ ɜɡɚɢɦɧɨ ɭɧɢɱɬɨɠɚɸɬɫɹ. ȿɫɥɢ ɜ ɩɨɫɥɟɞɧɟɣ ɫɤɨɛɤɟ ɫɬɨɢɬ ɩɪɨɫɬɨɟ ɱɢɫɥɨ 6l±1, ɬɨ ɧɚɲɟ ɩɪɟɞɩɨɥɨɠɟɧɢɟ ɜɟɪɧɨɟ. ȿɫɥɢ ɠɟ ɷɬɨ ɱɢɫɥɨ ɫɨɫɬɚɜɧɨɟ, ɬɨ ɩɨɫɬɭɩɢɦ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ. Ȼɭɞɟɦ ɨɬɧɢɦɚɬɶ ɨɬ ɱɢɫɥɚ 6l±1 ɱɢɫɥɚ, ɤɪɚɬɧɵɟ 6 (ɬɨ ɟɫɬɶ, 6, 12, 18 ɢ ɬɚɤ ɞɚɥɟɟ, ɨɛɨɡɧɚɱɢɦ ɢɯ ɤɚɤ 6z), ɩɨɤɚ ɜ ɢɬɨɝɟ ɧɟ ɩɨɥɭɱɢɦ ɩɪɨɫɬɨɟ ɱɢɫɥɨ. Ʉɚɤ ɢɡɜɟɫɬɧɨ, ɩɪɢ ɭɫɥɨɜɢɢ

6l±1–6z <25,

ɷɬɨ ɱɢɫɥɨ ɨɛɹɡɚɬɟɥɶɧɨ ɛɭɞɟɬ ɩɪɨɫɬɵɦ. ɩɪɢɱёɦ N>13, ɚ k, l, m –ɧɟɤɨɬɨɪɵɟ ɧɚɬɭɪɚɥɶɧɵɟ

N= (6k +6m+6z) + (6l±1–6z),

ɜ ɤɨɬɨɪɨɦ ɩɟɪɜɨɟ ɫɥɚɝɚɟɦɨɟ, ɨɱɟɜɢɞɧɨ, ɞɟɥɢɬɫɹ ɧɚɰɟɥɨ ɧɚ 6, ɚ ɜɬɨɪɨɟ ɫɥɚɝɚɟɦɨɟ –ɩɪɨɫɬɨɟ ɱɢɫɥɨ.

Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɢɡ ɜɵɲɟɢɡɥɨɠɟɧɧɨɝɨ ɦɨɠɧɨ ɫɞɟɥɚɬɶ ɜɵɜɨɞ, ɱɬɨ ɥɸɛɨɟ ɧɟɱёɬɧɨɟ ɱɢɫɥɨ, ɧɚɱɢɧɚɹ ɫ 3, ɦɨɠɧɨ ɩɪɟɞɫɬɚɜɢɬɶ ɤɚɤ ɫɭɦɦɭ ɞɜɭɯ ɱɢɫɟɥ, ɨɞɧɨ ɢɡ ɤɨɬɨɪɵɯ ɩɪɨɫɬɨɟ, ɚ ɜɬɨɪɨɟ ɤɪɚɬɧɨ ɱɢɫɥɭ 6. ɂɡɜɟɫɬɧɨ, ɱɬɨ ɥɸɛɨɟ ɱёɬɧɨɟ ɱɢɫɥɨ ɦɨɠɧɨ ɜɵɪɚɡɢɬɶ ɜ ɜɢɞɟ 6k+2 ɢɥɢ 6k+4 ɢɥɢ 6k+6. Ⱦɨɛɚɜɥɹɹ ɤ ɱёɬɧɨɦɭ ɱɢɫɥɭ 1, ɩɨɥɭɱɚɟɦ ɧɟɱёɬɧɨɟ ɱɢɫɥɨ ɢ, ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɷɬɨ ɧɟɱёɬɧɨɟ ɱɢɫɥɨ Nɦɨɠɟɬ ɢɦɟɬɶ ɜɢɞ

ɇɟɱёɬɧɨɟ ɱɢɫɥɨ

3 5 7 9 11 13 25

49 N=6k +3, N=6k +5, N=6k +7.

ɂɧɚɱɟ ɝɨɜɨɪɹ, ɧɟɱёɬɧɨɟ ɱɢɫɥɨ ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɫɭɦɦɭ ɩɪɨɫɬɨɝɨ ɱɢɫɥɚ (3, 5, 7) ɢ ɱɢɫɥɚ, ɤɪɚɬɧɨɝɨ 6.

Ɍɚɛɥɢɰɚ 1.

ȼɚɪɢɚɧɬɵ ɪɚɡɛɢɟɧɢɣ ɧɟɱёɬɧɵɯ ɱɢɫɟɥ Ɋɚɡɛɢɟɧɢɟ ɩɨ ɝɢɩɨɬɟɡɟ Ƚɨɥɶɞɛɚɯɚ

- - 3+2+2 3+3+3 5+3+3 7+3+3 7+5+13 7+7+11 7+5+37 7+11+31

ɉɪɟɞɥɚɝɚɟɦɨɟ ɪɚɡɛɢɟɧɢɟ

3+6∙0 5+6∙0 7+6∙0 3+6∙1 5+6∙1 7+6∙1 7+6∙3 13+6∙2 7+6∙7 19+6∙5

Ɋɚɡɛɢɟɧɢɟ ɱɟɪɟɡ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɵɟ ɱɢɫɥɚ

- - - 3+1+5 - -

7+1+3+5+9

7+1+3+5+9+11+13

ɂɡ ɥɢɬɟɪɚɬɭɪɵ ɢɡɜɟɫɬɧɨ, ɱɬɨ ɧɟɱёɬɧɨɟ ɱɢɫɥɨ ɦɨɠɧɨ ɩɪɟɞɫɬɚɜɢɬɶ ɤɚɤ ɫɭɦɦɭ ɬɪёɯ ɩɪɨɫɬɵɯ ɱɢɫɟɥ (ɞɨɤɚɡɚɧɧɚɹ ɬɟɪɧɚɪɧɚɹ ɝɢɩɨɬɟɡɚ Ƚɨɥɶɞɛɚɯɚ [3]). Ɍɚɤ ɠɟ ɧɟɱёɬɧɵɟ ɱɢɫɥɚ, ɤɨɬɨɪɵɟ ɹɜɥɹɸɬɫɹ ɩɨɥɧɵɦɢ ɤɜɚɞɪɚɬɚɦɢ (ɧɚɩɪɢɦɟɪ, ɱɢɫɥɚ 25, 49), ɦɨɝɭɬ ɛɵɬɶ ɡɚɩɢɫɚɧɵ ɤɚɤ ɫɭɦɦɚ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɵɯ ɧɟɱёɬɧɵɯ ɱɢɫɟɥ. ȼ ɬɚɛɥɢɰɟ 1 ɞɥɹ ɫɪɚɜɧɟɧɢɹ ɩɪɢɜɟɞɟɧɵ ɜɚɪɢɚɧɬɵ ɪɚɡɛɢɟɧɢɹ ɧɟɤɨɬɨɪɵɯ ɧɟɱёɬɧɵɯ ɱɢɫɟɥ ɧɚ ɫɨɫɬɚɜɧɵɟ ɱɚɫɬɢ. ɋɨɜɩɚɞɚɸɳɢɟ ɱɢɫɥɚ ɜɵɞɟɥɟɧɵ ɰɜɟɬɨɦ.

Ɇɨɠɧɨ ɝɪɚɮɢɱɟɫɤɢ ɢɡɨɛɪɚɡɢɬɶ ɜ ɞɜɭɦɟɪɧɵɯ ɞɟɤɚɪɬɨɜɵɯ ɤɨɨɪɞɢɧɚɬɚɯ ɫɜɹɡɶ ɦɟɠɞɭ ɜɟɥɢɱɢɧɨɣ ɧɟɱёɬɧɨɝɨ ɱɢɫɥɚ N ɢ ɱɢɫɥɨɦ k, ɢɫɯɨɞɹ ɢɡ ɡɚɩɢɫɚɧɧɵɯ ɜɵɲɟ ɬɪёɯ ɭɪɚɜɧɟɧɢɣ. Ɉɱɟɜɢɞɧɨ, ɱɬɨ ɷɬɨ ɛɭɞɭɬ ɬɪɢ ɩɪɹɦɵɟ ɥɢɧɢɢ, ɩɚɪɚɥɥɟɥɶɧɵɟ ɞɪɭɝ ɞɪɭɝɭ (ɪɢɫɭɧɨɤ 1). ɍɪɚɜɧɟɧɢɟ ɤɚɠɞɨɣ ɩɪɹɦɨɣ ɧɚ ɝɪɚɮɢɤɟ ɛɭɞɟɬ ɢɦɟɬɶ ɜɢɞ N=6k+ɪ,

ɝɞɟ ɱɢɫɥɨ ɪ –ɩɪɨɫɬɨɟ. Ɍɚɧɝɟɧɫ ɭɝɥɚ ɧɚɤɥɨɧɚ ɤɚɠɞɨɣ ɩɪɹɦɨɣ, ɨɱɟɜɢɞɧɨ, ɛɭɞɟɬ ɪɚɜɟɧ 6. Ɉɬɪɟɡɤɢ, ɨɬɫɟɤɚɟɦɵɟ ɩɪɹɦɵɦɢ ɩɨ ɜɟɪɬɢɤɚɥɶɧɨɣ ɤɨɨɪɞɢɧɚɬɧɨɣ ɩɪɹɦɨɣ, ɤɚɤ ɪɚɡ ɢ ɛɭɞɭɬ ɪɚɜɧɵ ɩɪɨɫɬɵɦ ɱɢɫɥɚɦ.

Ɂɚɦɟɬɢɦ, ɱɬɨ, ɫɞɜɢɝɚɹ ɞɜɟ ɜɟɪɯɧɢɟ ɩɪɹɦɵɟ ɥɢɧɢɢ, ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɫɥɭɱɚɹɦ ɪ=5 ɢ ɪ=7, ɩɨ ɜɟɪɬɢɤɚɥɢ ɜɜɟɪɯ, ɦɵ ɦɨɠɟɦ ɩɨɥɭɱɢɬɶ ɢ ɞɪɭɝɢɟ ɪɚɡɛɢɟɧɢɹ ɞɨɫɬɚɬɨɱɧɨ ɛɨɥɶɲɨɝɨ ɧɟɱёɬɧɨɝɨ ɱɢɫɥɚ N ɧɚ ɞɜɚ ɫɥɚɝɚɟɦɵɯ, ɨɞɧɨ ɢɡ ɤɨɬɨɪɵɯ ɛɭɞɟɬ ɤɪɚɬɧɨ 6, ɚ ɞɪɭɝɨɟ ɩɪɟɞɫɬɚɜɥɹɬɶ ɫɨɛɨɣ ɩɪɨɫɬɨɟ ɱɢɫɥɨ. ȼɵɜɨɞ. ɉɨɤɚɡɚɧɨ, ɱɬɨ ɥɸɛɨɟ ɧɟɱёɬɧɨɟ ɱɢɫɥɨ, ɧɚɱɢɧɚɹ ɫ 3, ɦɨɠɧɨ ɩɪɟɞɫɬɚɜɢɬɶ ɤɚɤ ɫɭɦɦɭ ɩɪɨɫɬɨɝɨ ɱɢɫɥɚ ɢ ɱɢɫɥɚ, ɤɪɚɬɧɨɝɨ 6.

Ɋɢɫɭɧɨɤ 1. Ƚɪɚɮɢɱɟɫɤɨɟ ɩɪɟɞɫɬɚɜɥɟɧɢɟ ɪɚɡɛɢɟɧɢɣ ɧɟɱёɬɧɨɝɨ ɱɢɫɥɚ

ɋɩɢɫɨɤ ɥɢɬɟɪɚɬɭɪɵ

1. Ⱥɦɨɫɨɜ ȿ.Ⱥ. Ɉ ɪɚɡɥɨɠɟɧɢɢ ɱёɬɧɵɯ ɱɢɫɟɥ ɧɚ ɩɪɨɫɬɵɟ ɫɥɚɝɚɟɦɵɟ // ɂɧɬɟɪɧɚɭɤɚ. – 2020. –№26(155). –ɑ.1. –ɋ.23-24. 2. Ⱥɦɨɫɨɜ ȿ.Ⱥ. Ɉ ɪɚɡɥɨɠɟɧɢɢ ɧɟɱёɬɧɨɝɨ ɱɢɫɥɚ ɧɚ ɩɪɨɫɬɵɟ ɫɥɚɝɚɟɦɵɟ // ɂɧɬɟɪɧɚɭɤɚ. – 2020. –№28(157). –ɋ.21-22. 3. Helfgott ɇ. Major arcs for Goldbach’s theorem // arxiv 1305.2897. 4. ɗɧɞɪɸɫ Ƚ. Ɍɟɨɪɢɹ ɪɚɡɛɢɟɧɢɣ. —Ɇ.: ɇɚɭɤɚ, 1982. —255 ɫ.